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 \(pq = 1\).