Matrices - Unit 7 HPreCalc

Inverse of a 2 by 2

Reciprocal of the determinate times matrix w/ d and a swapped and b and c with negative

AX = C → X = A^-1 C. ORDER MATTERS

As long as there’s a determinate, there’s an inverse

Add matrices

[2 1] + [4 3] = [6 4]

lit js add terms. Must hv same dimensions to do so

Also 3[2 1] can be rewritten as [6 2] js multiply out

Multiply matrices

It’s gotta be the column of the first matrices is same dimension of row as other.

• So you can do a 1 × 3 by a 3 × 4 , but not a 1 × 3 by a 4 × 3.

• Inside dimensions must match

The outside dimension is the answers dimensions, so in the example above, a 1 by 4.

• solving for C11: row 1 with column 1. 1A times 1A + 1B times 2A etc. 21 would be row 2 of 1st w/ column 1 of 2nd.

• The order you multiply MATTERS. If you change the order the answers different.

Matrix parts

Matrix A is a 2 by 2

Elements A11, A12, A21, A22

Matrix is written ROW BY COLUMN

A matrix is like a table.

Determinate of a 2 by 2

ad-bc

Determinate of a 3 by 3

a (ei - fh) -b (di - fg) + c(dh - eg)

shortcut(risky)

• sum of products of right diagonals minus sum of products of left diagonals

Area of a triangle

Make a 3 by 3 w/ column 1 as x column 2 as y column 3 all as 1. That determinant times ½ is the area.

Co-linear points

Same 3 by 3 as a triangle. If the determinate is 0 (so the area is 0) they are collinear

How to transform matrices

It would be an b x 2 by a 2 × 2, so it works column each for x and y

• Reflect along y axis (invert): [-1 0, 0 1]

• X axis (upside down): [1 0, 0 -1]

• Sign swap for diff axis x starts positive tho

• Multiply this matrix by the points matrix to transform all the points

Cramer’s rule on a 3 variable system

1. Make a matrix for the x y and z side. Find that determinant D

2. Replace the x with what each equation equals

   • Like if 2x + 3y + 4z = 6. [2 3 4] → [6 3 4]. Do this for all equations to make a matrix to solve for Dx

3. Repeat this but for y and z.

4. Dx/d is x coordinate and so on for y and z

If it’s a system to rewrite as a matrix:

{ax + by = c} → [a b : c] but it’s 3 dots

What does it mean when | x | instead of [ x ]

Ask Nikhil fr