Compute the product of the two matrices below, $AB$. Do this using the definitions of the matrix-vector product (Definition MVP) and the definition of matrix multiplication (Definition MM). \begin{align*} A= \begin{bmatrix} 2&5\\ -1&3\\ 2&-2 \end{bmatrix} && B=\begin{bmatrix} 1&5&-3&4\\ 2&0&2&-3 \end{bmatrix} \end{align*}

Compute the product $AB$ of the two matrices below using both the definition of the matrix-vector product (Definition MVP) and the definition of matrix multiplication (Definition MM). \begin{align*} A &= \begin{bmatrix} 1 & 3 & 2 \\ -1 & 2 & 1 \\ 0 & 1 & 0 \end{bmatrix} & B &= \begin{bmatrix} 4 & 1 & 2\\ 1 & 0 & 1\\3 & 1 & 5 \end{bmatrix} \end{align*}

Compute the product $AB$ of the two matrices below using both the definition of the matrix-vector product (Definition MVP) and the definition of matrix multiplication (Definition MM). \begin{align*} A&= \begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix} & B&= \begin{bmatrix} 2 & 3 \\ 4 & 6 \end{bmatrix} \end{align*}

Compute the product $AB$ of the two matrices below using both the definition of the matrix-vector product (Definition MVP) and the definition of matrix multiplication (Definition MM). \begin{align*} A&= \begin{bmatrix} 3 & 1 \\ 2 & 4 \\ 6 & 5\\1 & 2 \end{bmatrix} & B&= \begin{bmatrix} -3 & 1 \\ 4 & 2 \end{bmatrix} \end{align*}

Compute the product $AB$ of the two matrices below. \begin{align*} A&= \begin{bmatrix} 1 & 2 & 3 & -2 \\ 0 & 1 & -2 & -1\\ 1 & 1 & 3 & 1 \end{bmatrix} & B&= \begin{bmatrix} 3\\ 4 \\ 0 \\ 2 \end{bmatrix} \end{align*}

Compute the product $AB$ of the two matrices below. \begin{align*} A&= \begin{bmatrix} 1 & 2 & 3 & -2 \\ 0 & 1 & -2 & -1\\ 1 & 1 & 3 & 1 \end{bmatrix} & B&= \begin{bmatrix} -7\\ 3 \\ 1 \\ 1 \end{bmatrix} \end{align*}

Compute the product $AB$ of the two matrices below using both the definition of the matrix-vector product (Definition MVP) and the definition of matrix multiplication (Definition MM). \begin{align*} A&= \begin{bmatrix} 1 & 3 & 1\\ 0 & 1 & 0\\1 & 1 & 2 \end{bmatrix} & B&= \begin{bmatrix} 2 & -5 & -1 \\ 0 & 1 & 0\\-1 & 2 & 1 \end{bmatrix} \end{align*}

For the matrix $A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}$, find $A^2$, $A^3$, $A^4$. Find a general formula for $A^n$ for any positive integer $n$.

For the matrix $A = \begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}$, find $A^2$, $A^3$, $A^4$. Find a general formula for $A^n$ for any positive integer $n$.

For the matrix $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$, find $A^2$, $A^3$, $A^4$. Find a general formula for $A^n$ for any positive integer $n$.

For the matrix $A = \begin{bmatrix} 0 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$, find $A^2$, $A^3$, $A^4$. Find a general formula for $A^n$ for any positive integer $n$.

Suppose that $A$ is a square matrix and there is a vector, $\vect{b}$, such that $\linearsystem{A}{\vect{b}}$ has a unique solution. Prove that $A$ is nonsingular. Give a direct proof (perhaps appealing to Theorem PSPHS) rather than just negating a sentence from the text discussing a similar situation.

The conclusion of Theorem HMIP is $\innerproduct{A\vect{x}}{\vect{y}}=\innerproduct{\vect{x}}{\adjoint{A}\vect{y}}$. Use the same hypotheses, and prove the similar conclusion: $\innerproduct{\vect{x}}{A\vect{y}}=\innerproduct{\adjoint{A}\vect{x}}{\vect{y}}$. Two different approaches can be based on an application of Theorem HMIP. The first uses Theorem AA, while a second uses Theorem IPAC. Can you provide two proofs?

Prove the second part of Theorem MMZM.

Prove the second part of Theorem MMIM.

Prove the second part of Theorem MMDAA.

Prove the second part of Theorem MMSMM.

Suppose that $A$ is an $m\times n$ matrix and $\vect{x},\,\vect{y}\in\nsp{A}$. Prove that $\vect{x}+\vect{y}\in\nsp{A}$.

Suppose that $A$ is an $m\times n$ matrix, $\alpha\in\complex{\null}$, and $\vect{x}\in\nsp{A}$. Prove that $\alpha\vect{x}\in\nsp{A}$.

Suppose that $A$ is an $n\times n$ matrix. Prove that $\adjoint{A}A$ and $A\adjoint{A}$ are Hermitian matrices.

Suppose that $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix. Prove that the null space of $B$ is a subset of the null space of $AB$, that is $\nsp{B}\subseteq\nsp{AB}$. Provide an example where the opposite is false, in other words give an example where $\nsp{AB}\not\subseteq\nsp{B}$.

Suppose that $A$ is an $n\times n$ nonsingular matrix and $B$ is an $n\times p$ matrix. Prove that the null space of $B$ is equal to the null space of $AB$, that is $\nsp{B}=\nsp{AB}$. (Compare with Exercise MM.T40.)

Suppose $\vect{u}$ and $\vect{v}$ are any two solutions of the linear system $\linearsystem{A}{\vect{b}}$. Prove that $\vect{u}-\vect{v}$ is an element of the null space of $A$, that is, $\vect{u}-\vect{v}\in\nsp{A}$.

Give a new proof of Theorem PSPHS replacing applications of Theorem SLSLC with matrix-vector products (Theorem SLEMM).

Suppose that $\vect{x},\,\vect{y}\in\complex{n}$, $\vect{b}\in\complex{m}$ and $A$ is an $m\times n$ matrix. If $\vect{x}$, $\vect{y}$ and $\vect{x}+\vect{y}$ are each a solution to the linear system $\linearsystem{A}{\vect{b}}$, what can you say that is interesting about $\vect{b}$? Form an implication with the existence of the three solutions as the hypothesis and an interesting statement about $\linearsystem{A}{\vect{b}}$ as the conclusion, and then give a proof.