Matrices

What is a matrix

A rectangular array of numbers (called elements) arranged in rows and columns.

How are rows and columns defined in a matrix

Rows are horizontal arrays of numbers; columns are vertical arrays of numbers.

How is the order (dimensions) of a matrix written

As (number of rows) × (number of columns), e.g. a matrix with 2 rows and 3 columns has dim = 2 × 3.

What is a square matrix

A matrix that has the same number of rows as columns.

What is an identity matrix

A square matrix whose main diagonal elements are all 1 and all other elements are 0, denoted ( I ).

What is the 2×2 identity matrix

( I_2 = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} )

How is a general matrix element denoted, and what does (a_{23}) mean

Elements are denoted (a_{ij}), where (i) is the row number and (j) is the column number; (a_{23}) is the element in row 2, column 3.

What must be true for two matrices to be equal

They must have the same dimensions (order), and every corresponding element must be equal.

What is a null (zero) matrix

A matrix in which every element is zero, e.g. the 2×2 null matrix is ( \begin{pmatrix} 0 & 0 \ 0 & 0 \end{pmatrix} ).

What is the geometric interpretation of adding a column vector ( \begin{pmatrix} h \ k \end{pmatrix} ) to a point's position vector ( \begin{pmatrix} x \ y \end{pmatrix} )

It represents a translation of the point by (h) units horizontally and (k) units vertically.

Why does adding a 2×2 matrix to a 2×1 matrix "make no sense"

Matrices can only be added or subtracted if they have exactly the same dimensions (same number of rows and columns), since addition/subtraction is done element-by-element.

How are two matrices of the same order added or subtracted

Corresponding elements are added or subtracted individually.

Is matrix addition commutative

Yes: ( A + B = B + A ).

Is matrix addition associative

Yes: ( (A+B)+C = A+(B+C) ).

Is matrix subtraction commutative

No: ( A - B \neq B - A ) in general (in fact (A-B = -(B-A))).

Is matrix subtraction associative

No: ( (A-B)-C \neq A-(B-C) ) in general.

Is it true that (A + 0 = A) for the null matrix (0)

Yes, adding the null (zero) matrix leaves any matrix unchanged.

What is the geometric interpretation of multiplying a position vector ( \begin{pmatrix} x \ y \end{pmatrix} ) by a scalar (k)

It scales (dilates) the position vector by factor (k) from the origin, along the same line through the origin.

How do you multiply a matrix by a scalar

Multiply every element of the matrix by that scalar.

Is it true that (nA + mA = (n+m)A) for scalars (n, m)

Yes, scalar multiplication distributes over scalar addition.

If (A + X = B), what is (X)

(X = B - A)

Is it true that (k(A+B) = kA + kB) for scalar (k)

Yes, scalar multiplication distributes over matrix addition.

Describe the process of matrix multiplication for (AB). Each element in row (i) of (A) is multiplied by the corresponding element in column (j) of (B), and the products are summed; this gives the element in row (i), column (j) of the product (AB).

What condition must hold for two matrices (A) (order (m \times n)) and (B) (order (p \times q)) to be multiplied as (AB)

The number of columns of (A) must equal the number of rows of (B) (i.e. (n = p)); the matrices are then said to be conformable for multiplication.

If (A) is (m \times n) and (B) is (n \times q), what is the order of (AB)

(m \times q) (rows of (A) by columns of (B)).

Is matrix multiplication commutative in general

No, in general (AB \neq BA); order matters when multiplying matrices.

Can (AB) be defined while (BA) is undefined

Yes — this happens when the dimensions are conformable one way (columns of (A) = rows of (B)) but not the other way around.

Does (A^2 = 0) always imply (A = 0) for matrices

No, unlike ordinary numbers, a non-zero matrix can satisfy (A^2 = 0) (a "nilpotent" matrix), because matrix multiplication has no inverse in general and zero-divisors can exist.

For conformable matrices, is (A(B+C) = AB + AC) true

Yes, matrix multiplication is distributive over addition (left distributive law).

For conformable matrices, is (AX + BX = (A+B)X) true

Yes, this is the right distributive law: (AX+BX=(A+B)X).

For conformable matrices, is (XA + XB = X(A+B)) true

Yes, this is the left distributive law: (XA+XB=X(A+B)).

Why is it not generally true that ((A+B)^2 = A^2 + 2AB + B^2) for matrices

Because matrix multiplication is not commutative, so ((A+B)^2 = A^2+AB+BA+B^2), and since (AB \neq BA) in general, this cannot always simplify to (A^2+2AB+B^2).

What condition on (A) and (B) makes ((A+B)^2 = A^2+2AB+B^2) true

Only when (AB = BA) (i.e. (A) and (B) commute).

If two square matrices (A) and (B) satisfy (AB = BA = I), what is the relationship between them

(B) is the multiplicative inverse of (A) (written (B = A^{-1})), and (A) is the inverse of (B) (written (A = B^{-1})); note (A^{-1}) is NOT the reciprocal (1/A).

Can a square matrix be its own inverse (i.e. (A^{-1} = A))

Yes, this is possible for certain matrices, e.g. if (A^2 = I) (an "involutory" matrix), such as a reflection matrix.

What is the determinant of a 2×2 matrix (A = \begin{pmatrix} a & b \ c & d \end{pmatrix})

( \det(A) = |A| = ad - bc )

What does it mean for a matrix to be "invertible" or "non-singular"

Its inverse exists, which requires its determinant to be non-zero ((\det(A) \neq 0)).

What does it mean for a matrix to be "singular"

Its inverse does NOT exist, which occurs when its determinant equals zero ((\det(A) = 0)).

What is the formula for the inverse of a 2×2 matrix (A = \begin{pmatrix} a & b \ c & d \end{pmatrix})

[ A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \ -c & a \end{pmatrix} ] provided (ad - bc \neq 0).

What are the preset steps to find the inverse of a 2×2 matrix (A=\begin{pmatrix}a&b\c&d\end{pmatrix})

1. Compute (\det(A) = ad-bc).
   2. Check (\det(A) \neq 0) (otherwise no inverse exists).
   3. Swap the leading diagonal elements (a) and (d).
   4. Negate the off-diagonal elements (b) and (c).
   5. Multiply the resulting matrix by (\dfrac{1}{\det(A)}).

Given (AX = B) where (A) is invertible, how do you solve for (X)

Multiply both sides on the left by (A^{-1}): (X = A^{-1}B) (note: must multiply on the left since matrix multiplication is not commutative).

Given (XA = B) where (A) is invertible, how do you solve for (X)

Multiply both sides on the right by (A^{-1}): (X = BA^{-1}) (must multiply on the right this time).

Given the matrix equation ( \begin{pmatrix} 2 & 3 \ 1 & 4 \end{pmatrix} X = \begin{pmatrix} 5 \ 19 \end{pmatrix} ), what are the preset steps to solve for (X)

1. Identify (A=\begin{pmatrix}2&3\1&4\end{pmatrix}) and (B=\begin{pmatrix}5\19\end{pmatrix}).
   2. Find (\det(A)) and check it is non-zero.
   3. Compute (A^{-1}) using the 2×2 inverse formula.
   4. Compute (X = A^{-1}B) by matrix multiplication.

If (A), (B), (C) are non-singular and (AB = BC), how do you show (A^2 = BC^2B^{-1})

From (AB=BC), multiply both sides on the right by (B^{-1}): (A = BCB^{-1}); then (A^2 = (BCB^{-1})(BCB^{-1}) = BC(B^{-1}B)CB^{-1} = BC^2B^{-1}).

Is it true that (AX = BX \Rightarrow A = B) for non-zero conformable matrices (A, B, X)

Not necessarily true — this only follows if (X) is invertible (multiply both sides on the right by (X^{-1})); if (X) is singular, (A) need not equal (B).

Is it true that (A^{-1} = A) implies anything special about matrix (A)

It means (A^2 = I); such matrices include the identity matrix, reflection matrices, and certain other involutory matrices.

If (AB = BA = kI) for scalar (k), how do you express (A^{-1}) in terms of (B)

( A^{-1} = \dfrac{1}{k}B ), since (A \cdot \left(\dfrac{1}{k}B\right) = \dfrac{1}{k}(AB) = \dfrac{1}{k}(kI) = I).

How do you interpret a system of two linear equations in matrix form (AX = B)

The coefficients of the variables form matrix (A), the variables form column vector (X), and the constants on the right-hand side form column vector (B); e.g. (2x+3y=19), (5x+4y=3) becomes (\begin{pmatrix}2&3\5&4\end{pmatrix}\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}19\3\end{pmatrix}).

What are the preset steps to solve a system of two linear equations using the matrix inversion method

1. Write the system in matrix form (AX=B).
   2. Find (A^{-1}) (requires (\det(A)\neq 0)).
   3. Compute (X = A^{-1}B).
   4. Read off the values of the variables from (X).

How can you modify a consistent system of two linear equations to make it inconsistent (no solution)

Change one equation so its line becomes parallel (but not identical) to the other — i.e. make the coefficient matrix singular ((\det(A)=0)) while the constants don't match proportionally.

How can you modify a consistent system of two linear equations to make it redundant (infinitely many solutions)

Change one equation to be a scalar multiple of the other (the two lines become identical), making (\det(A)=0) with proportional constants too.

In an applied matrix problem where matrix (A) gives quantities (e.g. pens/pencils per pack) and matrix (B) gives unit costs, what does the product (AB) represent

The total cost of each pack (rows of (A) representing packs multiplied by the cost column vector (B) gives total cost per pack).

If (X = \begin{pmatrix}1 & 1\end{pmatrix}) is a row vector and (A) gives quantities of items per pack, what does (XA) represent

The total quantity of each item (e.g. total pens and total pencils) summed across all the packs.

What does (XAB) represent in an applied context (X = summing row vector, A = quantities, B = unit prices)

The total combined cost of all packs together.

Does the shape or size of an object change under a translation

No — translation preserves shape, size, orientation, and angles; every point moves by the same vector.

Is a translation a linear transformation

No, because a translation moves the origin away from itself (unless the translation vector is zero); true linear transformations must fix the origin.

Why does the origin always stay fixed under a linear transformation

Because for (x'=ax+by), (y'=cx+dy), substituting (x=0, y=0) gives (x'=0, y'=0); there are no constant terms added.

How do you express the linear transformation (x'=ax+by), (y'=cx+dy) in matrix form

[ \begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} ]

How do you represent the linear transformation (T: (x,y) \to (x-2y, 3x+4y)) as a matrix

( T = \begin{pmatrix} 1 & -2 \ 3 & 4 \end{pmatrix} ), since (x'=1\cdot x + (-2)\cdot y) and (y'=3\cdot x + 4\cdot y).

What are the preset steps to find the image of a point under a linear transformation matrix (T)

1. Write the point as a column vector (\begin{pmatrix}x\y\end{pmatrix}).
   2. Multiply (T) by the column vector: (T\begin{pmatrix}x\y\end{pmatrix}).
   3. The resulting column vector gives the image coordinates ((x', y')).

How do you find the transformation matrix (T) given a set of points and their images

Set up simultaneous equations using (x'=ax+by), (y'=cx+dy) for at least two known point/image pairs, then solve for (a, b, c, d); alternatively, form matrices (P) (object points as columns) and (P') (image points as columns) so that (TP=P'), then (T=P'P^{-1}).

What is the transformation matrix for a reflection about the x-axis

( \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix} ), since (x'=x) and (y'=-y).

What is the transformation matrix for a reflection about the y-axis

( \begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix} ), since (x'=-x) and (y'=y).

What is the transformation matrix for a reflection about the line (y=x)

( \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix} ), since (x'=y) and (y'=x) (coordinates are swapped).

What is the transformation matrix for a reflection about the line (y=-x)

( \begin{pmatrix} 0 & -1 \ -1 & 0 \end{pmatrix} ), since (x'=-y) and (y'=-x).

What is the general transformation matrix for a reflection about a line through the origin inclined at angle (\theta) to the x-axis

[ T = \begin{pmatrix} \cos 2\theta & \sin 2\theta \ \sin 2\theta & -\cos 2\theta \end{pmatrix} ]

Given (\cos 2\theta = \dfrac{1-m^2}{1+m^2}) and (\sin 2\theta = \dfrac{2m}{1+m^2}) where (m=\tan\theta), what is this used for

To find the reflection matrix about a line (y=mx) through the origin, using the gradient (m) directly instead of the angle (\theta).

What are the preset steps to find the reflection matrix about a line (y=mx) using its gradient (m)

1. Compute (\cos 2\theta = \dfrac{1-m^2}{1+m^2}) and (\sin 2\theta = \dfrac{2m}{1+m^2}).
   2. Substitute into (T = \begin{pmatrix}\cos2\theta & \sin2\theta \ \sin2\theta & -\cos2\theta\end{pmatrix}).

What is the transformation matrix for a rotation of angle (\theta) anti-clockwise about the origin

[ T = \begin{pmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{pmatrix} ]

What are the preset steps to derive the anticlockwise rotation matrix by angle (\theta)

1. Note the image of (A(1,0)) under rotation (\theta) anticlockwise is ((\cos\theta, \sin\theta)).
   2. Note the image of (B(0,1)) (found by rotating (A) a further 90°) is ((-\sin\theta, \cos\theta)).
   3. Place these images as the columns of the transformation matrix: (T=\begin{pmatrix}\cos\theta & -\sin\theta \ \sin\theta & \cos\theta\end{pmatrix}).

What is the transformation matrix for a rotation of angle (\theta) clockwise about the origin

[ T = \begin{pmatrix} \cos\theta & \sin\theta \ -\sin\theta & \cos\theta \end{pmatrix} ] (equivalent to substituting (-\theta) into the anticlockwise rotation matrix).

What is the transformation matrix for a 90° anti-clockwise rotation about the origin

( \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} ), since (x'=-y) and (y'=x).

What is the transformation matrix for a 90° clockwise rotation about the origin

( \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} ), since (x'=y) and (y'=-x).

What is the transformation matrix for a 180° rotation about the origin (either direction)

( \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} ), since (x'=-x) and (y'=-y).

Is a 180° rotation about the origin the same as a reflection about (y=x)

No — a 180° rotation sends ((x,y)\to(-x,-y)), while reflection in (y=x) sends ((x,y)\to(y,x)); these are different transformations (though both are involutory).

Is a 180° rotation about the origin the same as a reflection about (y=-x)

No — a 180° rotation sends ((x,y)\to(-x,-y)), while reflection in (y=-x) sends ((x,y)\to(-y,-x)); these are different in general.

How can a 180° anti-clockwise rotation be produced as a combination of two reflections

By reflecting about any two perpendicular lines through the origin (lines at 90° to each other), e.g. reflection about the x-axis followed by reflection about the y-axis.

What relationship exists between the matrix (M) for a 90° clockwise rotation and (T) for a 90° anticlockwise rotation

(M = T^{-1} = T^3) (rotating 90° clockwise reverses a 90° anticlockwise rotation), and (M) is also equal to the matrix for a 270° anti-clockwise rotation, since rotating 270° anticlockwise gives the same result as rotating 90° clockwise.

What are the coordinates of the image of point (B(0,1)) under a rotation (\theta) anti-clockwise about the origin, given that (A(1,0) \to (\cos\theta, \sin\theta))

((-\sin\theta, \cos\theta)), since (B) is 90° ahead of (A) around the origin.

How can the matrix method for rotations be used to derive the double angle identities (\cos2\theta = \cos^2\theta - \sin^2\theta) and (\sin2\theta = 2\sin\theta\cos\theta)

By showing that a rotation of (2\theta) is the same as applying a rotation of (\theta) twice, i.e. (R(\theta)R(\theta) = R(2\theta)); multiplying (R(\theta)) by itself and comparing entries with (R(2\theta)) gives the double angle formulas.

What is the transformation matrix for a dilation parallel to the x-axis by factor (h) (y-coordinate unchanged)

( \begin{pmatrix} h & 0 \ 0 & 1 \end{pmatrix} ), since (x'=hx) and (y'=y).

What is the transformation matrix for a dilation parallel to the y-axis by factor (k) (x-coordinate unchanged)

( \begin{pmatrix} 1 & 0 \ 0 & k \end{pmatrix} ), since (x'=x) and (y'=ky).

What is the general transformation matrix for a dilation ((x,y)\to(\lambda_1 x, \lambda_2 y))

( \begin{pmatrix} \lambda_1 & 0 \ 0 & \lambda_2 \end{pmatrix} )

What happens to a shape under a dilation if the scale factor (h) or (k) is negative

The shape is also reflected in the axis (in addition to being scaled), since a negative factor reverses direction along that axis.

What transformation does the matrix ( \begin{pmatrix} 2 & 0 \ 0 & 3 \end{pmatrix} ) represent

A dilation (stretch) by factor 2 parallel to the x-axis combined with a dilation by factor 3 parallel to the y-axis.

What transformation does the matrix ( \begin{pmatrix} 2 & 0 \ 0 & 2 \end{pmatrix} ) represent

An enlargement (dilation) of scale factor 2 from the origin in all directions (a uniform scaling, not just along an axis).

Given a transformation matrix (T) and an image point ((x',y')), how do you find the original object point ((x,y))

Compute the inverse matrix (T^{-1}) and multiply it by the image column vector: (\begin{pmatrix}x\y\end{pmatrix} = T^{-1}\begin{pmatrix}x'\y'\end{pmatrix}).

Why does the matrix (S) which reverses the effect of transformation (T) equal (T^{-1})

Because applying (T) then (S) must return every point to itself, i.e. (ST = I), which by definition means (S = T^{-1}).

What is the matrix that reverses the effect of a dilation matrix ( \begin{pmatrix} 3 & 0 \ 0 & 3 \end{pmatrix} )

Its inverse, ( \begin{pmatrix} 1/3 & 0 \ 0 & 1/3 \end{pmatrix} ), which is a dilation by factor (1/3).

What matrix reverses the effect of a rotation (\theta) anti-clockwise about the origin, (T=\begin{pmatrix}\cos\theta & -\sin\theta \ \sin\theta & \cos\theta\end{pmatrix})

(T^{-1} = \begin{pmatrix}\cos\theta & \sin\theta \ -\sin\theta & \cos\theta\end{pmatrix}), which is the rotation matrix for angle (\theta) clockwise (equivalently, rotation (-\theta)).

Why does a reflection matrix reverse its own effect (i.e. (T^{-1}=T) for reflections)

Because reflecting a point twice about the same line returns it to its original position, so (T^2 = I), meaning (T) is its own inverse.

Give an example (besides reflections) of a transformation matrix (T) such that (T^{-1}=T). The identity matrix (I), or a 180° rotation matrix (\begin{pmatrix}-1&0\0&-1\end{pmatrix}) (since applying it twice returns each point to its original position).

How do you find a single matrix representing two combined transformations, first (S) then (T)

Multiply the matrices in the order (TS) (the matrix of the transformation applied second goes on the left) and apply this combined matrix to the object's coordinates.

Why does the order of matrix multiplication matter when combining two transformations (S) and (T)

Because matrix multiplication is not commutative in general ((ST \neq TS)), so performing (S) then (T) gives a different result than performing (T) then (S), unless the transformations happen to commute.

If (S=\begin{pmatrix}0&1\1&0\end{pmatrix}) (reflection in (y=x)), what type of matrix (T) would satisfy (ST=TS)

A matrix that commutes with (S), such as the identity matrix, a scalar multiple of the identity (uniform dilation), or (S) itself — generally matrices with symmetric structure relative to (S)'s axis of reflection.

What single transformation has the same combined effect as reflecting in the x-axis then reflecting in the y-axis (in either order)

A 180° rotation about the origin, represented by ( \begin{pmatrix} -1 & 0 \ 0 & -1 \end{pmatrix} ).

What is the general rule for the combined effect of two reflections about lines through the origin at angles (\alpha) and (\beta) to the x-axis (reflect in (y=\tan\alpha \, x) then (y=\tan\beta \, x))

The combined effect is a rotation of (\theta = 2(\beta-\alpha)) anti-clockwise about the origin.

What are the preset steps to prove that reflecting in (y=(\tan\alpha)x) then (y=(\tan\beta)x) equals a rotation of (2(\beta-\alpha)) anticlockwise

1. Write the reflection matrix for angle (\alpha): (R_\alpha = \begin{pmatrix}\cos2\alpha & \sin2\alpha \ \sin2\alpha & -\cos2\alpha\end{pmatrix}).
   2. Write the reflection matrix for angle (\beta) similarly.
   3. Multiply in order (R_\beta R_\alpha) (second transformation on the left).
   4. Expand using compound angle identities to show the product simplifies to the rotation matrix for angle (2(\beta-\alpha)).

If two reflections (y=(\tan40°)x) then (y=(\tan\theta)x) combine to give a 120° anti-clockwise rotation, how do you find (\theta)

Use (2(\theta - 40°) = 120°), so (\theta - 40° = 60°), giving (\theta = 100°).

What is the inverse of a combined transformation (ST) (i.e. (S) then (T))

((ST)^{-1} = T^{-1}S^{-1}) — the inverses are applied in reverse order.