2.1 Exercises

20 points · 4 extra credit

Matrix Operations

1. 1. (2 pts) Let

   $A = \begin{bmatrix} 1 & -2 & 3 \\ 4 & 0 & -1 \end{bmatrix}, \qquad B = \begin{bmatrix} 2 & 1 & 0 \\ -3 & 5 & 2 \end{bmatrix}.$

   Compute $2A - B$.

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2. 2. (3 pts) Let $A = \begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 4 \\ -2 & 5 \end{bmatrix}$. Compute both $AB$ and $BA$, and state what the results show about matrix multiplication.

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3. 3. (3 pts) Compute the product $AB$, where

   $A = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 2 & 1 \\ 0 & -1 \\ 4 & 2 \end{bmatrix}.$

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4. 4. (2 pts) Write down the transpose $A^{\mathsf T}$ of the matrix $A$ from Problem 1.

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5. 5. (3 pts) Let $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$. Compute $A^2$ and $A^3$, then conjecture a formula for $A^n$ ($n \ge 1$).

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6. 6. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) For all $n\times n$ matrices $A$ and $B$, $(A+B)^2 = A^2 + 2AB + B^2$.

   (b) If $A$ and $B$ are matrices with $AB = 0$, then $A = 0$ or $B = 0$.

   (c) The $(i,j)$ entry of $AB$ equals the sum of products of the entries of row $i$ of $A$ with the corresponding entries of column $j$ of $B$.

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7. 7. (2 pts) Let $\mathbf{u} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} 2 \\ -1 \end{bmatrix}$. Compute the outer product $\mathbf{u}\mathbf{v}^{\mathsf T}$ and state its size.

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8. 8. (2 pts) Let $A$ be any square matrix. Show that $A + A^{\mathsf T}$ is symmetric and that $A - A^{\mathsf T}$ is skew-symmetric (i.e., equal to the negative of its own transpose).

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9. 9. (4 pts ••) The trace of an $n\times n$ matrix $A$, written $\operatorname{tr}(A)$, is the sum of its diagonal entries $a_{11} + a_{22} + \cdots + a_{nn}$. Prove that for any $n\times n$ matrices $A$ and $B$,

   $\operatorname{tr}(AB) = \operatorname{tr}(BA),$

   even though $AB \ne BA$ in general.

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