W3 W4 Linear Algebra OSU

Invertible matrix theorem

1. A is invertible.

There exists C such that CA=I_n

Ax=0 has only the trivial solution.

Ahas n pivot positions.

A is row equivalent to I_n

There exists D such that AD=I_n

Ax=b has a solution for every b∈R^n

A^T is invertible

Is the inverse of an elementary matrix also elementary?

Yes, it corresponds to the inverse row operation.

How can you find the inverse of a matrix using row reduction?

Row reduce (A ∣ In)] to (I_n ∣ A−1)

What does it mean for two matrices to be row equivalent?

One can be obtained from the other using a sequence of row operations.

What must be true for a square matrix A to be invertible?

A must be row equivalent to In

What does the Invertible Matrix Theorem imply about solving Ax=b?

If A is invertible, then Ax=b has a unique solution for every b∈R^n

What is the determinant of a triangular matrix?

The product of the entries on the main diagonal.

How do you prove that the determinant of an upper-triangular matrix is the product of the diagonal entries?

By induction and cofactor expansion on the first column.

What happens to det⁡(A) if a row is multiplied by a scalar c?

det(B)=c⋅det(A)

What happens to det⁡(A) if two rows are switched?

det(B) = -det(A)

How can you use row operations to compute a determinant?

Reduce the matrix to triangular form, keeping track of scalar multiplications and row swaps.

What is the multiplicative property of determinants?

det(AB)=det(A)⋅det(B)

Why does the multiplicative property hold when A is not invertible?

Because A not invertible ⇒ det⁡(A)=0 ⇒ AB not invertible ⇒ det⁡(AB)=0

so both sides =0

What is the formula for the determinant of an elementary matrix multiplied by a matrix B?

det(EB)=det(E)⋅det(B)

What is Cramer's Rule for solving Ax=b?

If A is invertible, then the solution vector x has entries

x_i = (det(Ai(b)) / (det(A)), i = 1, 2, n...

What is the adjugate (adjoint) matrix of A?

The transpose of the cofactor matrix of A

What is the span of a single non-zero vector in R^n?

A line through the origin

True or False: If a set of vectors includes the zero vector, it is linearly independent.

False - its dependent

What does it mean if one vector is in the span of others?

The set of vectors is linearly dependent

What does the matrix

0 1

1 0

do to a vector

x

y

t swaps x and y; this reflects across the line y=x

What is the standard matrix of a rotation by θ?

cos θ -sin θ

sin θ cos θ

What is the matrix for projection onto the x-axis?

1 0

0 0

What is the standard matrix for projection onto vector

u = (a,b)^T?

1/(a^2 + b^2) * | a^2 ab

ab b^2

What is the matrix for composing two transformations T∘S where T(x)=Bxand S(x)=Ax?

The matrix is BA