2.2E: Matrix Addition, Scalar Multiplication, and Transposition Exercises

Exercises for 1

Exercise $\PageIndex{1}$

Find $a$, $b$, $c$, and $d$ if

1. $\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] = \left[ \begin{array}{cc} c - 3d & -d \\ 2a + d & a + b \end{array} \right]$
2. $\left[ \begin{array}{cc} a - b & b - c \\ c - d & d - a \end{array} \right] = 2 \left[ \begin{array}{rr} 1 & 1 \\ -3 & 1 \end{array} \right]$
3. $3 \left[ \begin{array}{c} a \\ b \end{array} \right] + 2 \left[ \begin{array}{rr} b \\ a \end{array} \right] = \left[ \begin{array}{rr} 1 \\ 2 \end{array} \right]$
4. $\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] = \left[ \begin{array}{cc} b & c \\ d & a \end{array} \right]$

Answer

2. $(a\ b\ c\ d) = (-2, -4, -6, 0) + t(1, 1, 1, 1)$,
3. $t$ arbitrary $a = b = c = d = t$, $t$ arbitrary

Exercise $\PageIndex{2}$

Compute the following:

1. $\left[\begin{array}{lll}3 & 2 & 1 \\5 & 1 & 0\end{array}\right]-5\left[\begin{array}{rrr}3 & 0 & -2 \\1 & -1 & 2\end{array}\right]$
2. $3\left[\begin{array}{r}3 \\-1\end{array}\right]-5\left[\begin{array}{l}6 \\2\end{array}\right]+7\left[\begin{array}{r}1 \\-1 \end{array}\right]$
3. $\left[\begin{array}{rr}-2 & 1 \\3 & 2\end{array}\right]-4\left[\begin{array}{ll}1 & -2 \\0 & -1\end{array}\right]+3\left[\begin{array}{rr}2 & -3 \\-1 & -2\end{array}\right]$
4. $\left[\begin{array}{lll}3 & -1 & 2\end{array}\right]-2\left[\begin{array}{lll}9 & 3 & 4\end{array}\right]+\left[\begin{array}{lll}3 & 11 & -6\end{array}\right]$
5. $\left[\begin{array}{rrrr}1 & -5 & 4 & 0 \\2 & 1 & 0 & 6\end{array}\right]^T$
6. $\left[\begin{array}{rrr}0 & -1 & 2 \\1 & 0 & -4 \\-2 & 4 & 0\end{array}\right]^T$
7. $\left[\begin{array}{rr}3 & -1 \\2 & 1\end{array}\right]-2\left[\begin{array}{rr}1 & -2 \\1 & 1\end{array}\right]^T$
8. $3\left[\begin{array}{rr}2 & 1 \\-1 & 0\end{array}\right]^T-2\left[\begin{array}{rr}1 & -1 \\2 & 3\end{array}\right]$

Answer

2. $\left[ \begin{array}{r} -14 \\ -20 \end{array} \right]$ $(-12, 4, -12)$
3. $\left[ \begin{array}{rrr} 0 & 1 & -2 \\ -1 & 0 & 4 \\ 2 & -4 & 0 \end{array} \right]$
4. $\left[ \begin{array}{rr} 4 & -1 \\ -1 & -6 \end{array} \right]$

Exercise $\PageIndex{3}$

Let $A = \left[ \begin{array}{rr} 2 & 1 \\ 0 & -1 \end{array} \right]$,
$B = \left[ \begin{array}{rrr} 3 & -1 & 2 \\ 0 & 1 & 4 \end{array} \right]$, $C = \left[ \begin{array}{rr} 3 & -1 \\ 2 & 0 \end{array} \right]$,
$D = \left[ \begin{array}{rr} 1 & 3 \\ -1 & 0 \\ 1 & 4 \end{array} \right]$, and $E = \left[ \begin{array}{rrr} 1 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right]$.

Compute the following (where possible).

1. $3A - 2B$
2. $5C$
3. $3E^{T}$
4. $B + D$
5. $4A^{T} - 3C$
6. $(A + C)^{T}$
7. $2B - 3E$
8. $A - D$
9. ((B - 2E)^{T}\)

Answer

2. $\left[ \begin{array}{rr} 15 & -5 \\ 10 & 0 \end{array} \right]$
3. Impossible
4. $\left[ \begin{array}{rr} 5 & 2 \\ 0 & -1 \end{array} \right]$
5. Impossible

Exercise $\PageIndex{4}$

Find $A$ if:

1. $5A - \left[ \begin{array}{rr} 1 & 0 \\ 2 & 3 \end{array} \right] = 3A - \left[ \begin{array}{rr} 5 & 2 \\ 6 & 1 \end{array} \right]$
2. $3A - \left[ \begin{array}{r} 2 \\ 1 \end{array} \right] = 5A - 2 \left[ \begin{array}{rr} 3 \\ 0 \end{array} \right]$

Answer

b. $\left[ \begin{array}{r} 2 \\ -\frac{1}{2} \end{array} \right]$

Exercise $\PageIndex{5}$

Find $A$ in terms of $B$ if:

1. $A + B = 3A + 2B$
2. $2A - B = 5(A + 2B)$

Answer

b. $A = -\frac{11}{3}B$

Exercise $\PageIndex{6}$

If $X$, $Y$, $A$, and $B$ are matrices of the same size, solve the following systems of equations to obtain $X$ and $Y$ in terms of $A$ and $B$.

1. $5X + 3Y = A \\ 2X + Y = B$
2. $4X + 3Y = A \\ 5X + 4Y = B$

Answer

b. $X = 4A - 3B$, $Y = 4B - 5A$

Example $\PageIndex{7}$

Find all matrices $X$ and $Y$ such that:

1. $3X - 2Y = \left[ \begin{array}{rr} 3 & - 1 \end{array} \right]$
2. $2X - 5Y = \left[ \begin{array}{rr} 1 & 2 \end{array} \right]$

Solution

$Y = (s, t)$, $X = \frac{1}{2}(1 + 5s, 2 + 5t)$; $s$ and $t$ arbitrary

Exercise $\PageIndex{8}$

Simplify the following expressions where $A$, $B$, and $C$ are matrices.

1. $2 \left[ 9(A - B) + 7(2B - A) \right]$
   $- 2 \left[ 3(2B + A) - 2(A + 3B) - 5(A + B) \right]$
2. $5 \left[ 3(A - B + 2C) - 2(3C - B) - A \right]$
   $+ 2 \left[ 3(3A - B + C) + 2(B - 2A) - 2C \right]$

Answer

b. $20A - 7B + 2C$

Exercise $\PageIndex{9}$

If $A$ is any $2 \times 2$ matrix, show that:

1. $A = a \left[ \begin{array}{rr} 1 & 0 \\ 0 & 0 \end{array} \right] + b \left[ \begin{array}{rr} 0 & 1 \\ 0 & 0 \end{array} \right] + c \left[ \begin{array}{rr} 0 & 0 \\ 1 & 0 \end{array} \right] + d \left[ \begin{array}{rr} 0 & 0 \\ 0 & 1 \end{array} \right]$ for some numbers $a$, $b$, $c$, and $d$.
2. $A = p \left[ \begin{array}{rr} 1 & 0 \\ 0 & 1 \end{array} \right] + q \left[ \begin{array}{rr} 1 & 1 \\ 0 & 0 \end{array} \right] + r \left[ \begin{array}{rr} 1 & 0 \\ 1 & 0 \end{array} \right] + s \left[ \begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array} \right]$ for some numbers $p$, $q$, $r$, and $s$.

Answer

b. If $A = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]$, then $(p, q, r, s) = \frac{1}{2}(2d, a + b - c - d, a - b + c - d, -a + b + c + d)$.

Exercise $\PageIndex{10}$

Let $A = \left[ \begin{array}{rrr} 1 & 1 & -1 \end{array} \right]$,
$B = \left[ \begin{array}{rrr} 0 & 1 & 2 \end{array} \right]$, and $C = \left[ \begin{array}{rrr} 3 & 0 & 1 \end{array} \right]$. If
$rA + sB + tC = 0$ for some scalars $r$, $s$, and $t$, show that necessarily $r = s = t = 0$.

Exercise $\PageIndex{11}$

1. If $Q + A = A$ holds for every $m \times n$ matrix $A$, show that $Q = 0_{mn}$.
2. If $A$ is an $m \times n$ matrix and $A + A^\prime = 0_{mn}$, show that \(A^\prime = -A\

Answer

b. If $A + A^\prime = 0$ then $-A = -A + 0 = -A + (A + A^\prime) = (-A + A) + A^\prime = 0 + A^\prime = A^\prime$

Exercise $\PageIndex{12}$

If $A$ denotes an $m \times n$ matrix, show that $A = -A$ if and only if $A = 0$.

Exercise $\PageIndex{13}$

A square matrix is called a diagonal matrix if all the entries off the main diagonal are zero. If $A$ and $B$ are diagonal matrices, show that the following matrices are also diagonal.

1. $A + B$
2. $A - B$
3. $kA$ for any number $k$

Answer

Write $A = \diag(a_{1}, \dots, a_{n})$, where $a_{1}, \dots, a_{n}$ are the main diagonal entries. If $B = \diag(b_{1}, \dots, b_{n})$ then $kA = \diag(ka_{1}, \dots, ka_{n})$.

Exercise $\PageIndex{14}$

In each case determine all $s$ and $t$ such that the given matrix is symmetric:

1. $\left[ \begin{array}{rr} 1 & s \\ -2 & t \end{array} \right]$
2. $\left[ \begin{array}{cc} s & t \\ st & 1 \end{array} \right]$
3. $\left[ \begin{array}{crc} s & 2s & st \\ t & -1 & s \\ t & s^{2} & s \end{array} \right]$
4. $\left[ \begin{array}{ccc} 2 & s & t \\ 2s & 0 & s + t \\ 3 & 3 & t \end{array} \right]$

Answer

2. $s = 1$ or $t = 0$
3. $s = 0$, and $t = 3$

Exercise $\PageIndex{15}$

In each case find the matrix $A$.

1. $\left(A + 3 \left[ \begin{array}{rrr} 1 & -1 & 0 \\ 1 & 2 & 4 \end{array} \right] \right)^{T} = \left[ \begin{array}{rr} 2 & 1 \\ 0 & 5 \\ 3 & 8 \end{array} \right]$
2. $\left(3A^{T} + 2 \left[ \begin{array}{rr} 1 & 0 \\ 0 & 2 \end{array} \right] \right)^{T} = \left[ \begin{array}{rr} 8 & 0 \\ 3 & 1 \end{array} \right]$
3. $\left(2A - 3 \left[ \begin{array}{rrr} 1 & 2 & 0 \end{array} \right] \right)^{T} = 3A^{T} + \left[ \begin{array}{rrr} 2 & 1 & -1 \end{array} \right]^{T}$
4. $\left(2A^{T} - 5 \left[ \begin{array}{rr} 1 & 0 \\ -1 & 2 \end{array} \right] \right)^{T} = 4A - 9 \left[ \begin{array}{rr} 1 & 1 \\ -1 & 0 \end{array} \right]$

Answer

2. $\left[ \begin{array}{rr} 2 & 0 \\ 1 & -1 \end{array} \right]$
3. $\left[ \begin{array}{rr} 2 & 7 \\ -\frac{9}{2} & -5 \end{array} \right]$

Exercise $\PageIndex{16}$

Let $A$ and $B$ be symmetric (of the same size). Show that each of the following is symmetric.

1. $(A - B)$
2. $kA$ for any scalar $k$

Answer

1. $A = A^{T}$,
2. $(kA)^{T} = kA^{T} = kA$.

Exercise $\PageIndex{17}$

Show that $A + A^{T}$ is symmetric for any square matrix $A$.

Exercise $\PageIndex{18}$

If $A$ is a square matrix and $A = kA^{T}$ where $k \neq \pm 1$, show that $A = 0$.

Exercise $\PageIndex{19}$

In each case either show that the statement is true or give an example showing it is false.

1. If $A + B = A + C$, then $B$ and $C$ have the same size.
2. If $A + B = 0$, then $B = 0$.
3. If the $(3, 1)$-entry of $A$ is $5$, then the $(1, 3)$-entry of $A^{T}$ is $-5$.
4. $A$ and $A^{T}$ have the same main diagonal for every matrix $A$.
5. If $B$ is symmetric and $A^{T} = 3B$, then $A = 3B$.
6. If $A$ and $B$ are symmetric, then $kA + mB$ is symmetric for any scalars $k$ and $m$.

Answer

1. False. Take $B = -A$ for any $A \neq 0$.
2. True. Transposing fixes the main diagonal.
3. True. $(kA + mB)^{T} = (kA)^{T} + (mB)^{T} = kA^{T} + mB^{T} = kA + mB$

Exercise $\PageIndex{20}$

A square matrix $W$ is called skew-symmetric if $W^{T} = -W$. Let $A$ be any square matrix.

1. Show that $A - A^{T}$ is skew-symmetric.
2. Find a symmetric matrix $S$ and a skew-symmetric matrix $W$ such that $A = S + W$.
3. Show that $S$ and $W$ in part (b) are uniquely determined by $A$.

Answer

c. Suppose $A = S + W$, where $S = S^{T}$ and $W = -W^{T}$. Then $A^{T} = S^{T} + W^{T} = S - W$, so $A + A^{T} = 2S$ and $A - A^{T} = 2W$. Hence $S = \frac{1}{2}(A + A^{T})$ and $W = \frac{1}{2}(A - A^{T})$ are uniquely determined by $A$.

Exercise $\PageIndex{21}$

If $W$ is skew-symmetric (Exercise $\PageIndex{20}$), show that the entries on the main diagonal are zero.

Exercise $\PageIndex{22}$

Prove the following parts of Theorem [thm:002170].

1. $(k + p)A = kA + pA$
2. $(kp)A = k(pA)$

Answer

b. If $A = \left[ a_{ij} \right]$ then $(kp)A = \left[ (kp)a_{ij} \right] = \left[ k(pa_{ij}) \right] = k \left[ pa_{ij} \right] = k(pA)$.

Exercise $\PageIndex{23}$

Let $A, A_{1}, A_{2}, \dots, A_{n}$ denote matrices of the same size. Use induction on $n$ to verify the following extensions of properties 5 and 6 of Theorem $\PageIndex{1}$.

1. (k(A_{1} + A_{2} + \dots + A_{n}) = kA_{1} + kA_{2} + \dots + kA_{n}\) for any number $k$
2. $(k_{1} + k_{2} + \dots + k_{n})A = k_{1}A + k_{2}A + \dots + k_{n}A$ for any numbers $k_{1}, k_{2}, \dots, k_{n}$

Example $\PageIndex{1}$

Let $A$ be a square matrix. If $A = pB^{T}$ and $B = qA^{T}$ for some matrix $B$ and numbers $p$ and $q$, show that either $A = 0 = B$ or $pq = 1$.