
# 12.A: Exercises for Chapter A

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## Calculational Exercises

1. In each of the following, ﬁnd matrices $$A, x,$$ and $$b$$ such that the given system of linear equations can be expressed as the single matrix equation $$Ax = b.$$

$(a)~~ \left. \begin{array}{ccccccc} 2x_1 &-& 3x_2& + &5x_3 &= &7 \\ 9x_1& - &x_2& +& x_3& =& -1 \\ x_1& + &5x_2& +& 4x_3 &= &0 \end{array} \right\} ~~~ (b)~~ \left. \begin{array}{ccccccccc} 4x_1&&& -& 3x_3& +& x_4& =& 1 \\ 5x_1& +& x_2&&& -& 8x_4& =& 3 \\ 2x_1& - &5x_2& + &9x_3& -& x_4& =& 0 \\ &&3x_2& - &x_3& +& 7x_4& =& 2\end{array} \right\}$

2. In each of the following, express the matrix equation as a system of linear equations.

$(a) \left[ \begin{array}{ccc} 3 & -1 & 2 \\ 4 & 3 & 7 \\ -2& 1 & 5 \end{array} \right] \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array} \right] = \left[ \begin{array}{c} 2 \\ -1 \\ 4 \end{array} \right] ~~~ (b)\left[ \begin{array}{cccc} 3 & -2 & 0&1 \\ 5 & 0 & 2 & -2\\ 3& 1 & 4&7\\ -2&5&1&6 \end{array} \right] \left[ \begin{array}{c} w \\ x\\ y\\ z \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0\\ 0\\ 0 \end{array} \right]$

3. Suppose that $$A, B, C, D,$$ and $$E$$ are matrices over $$\mathbb{F}$$ having the following sizes:

$A {\it{~is~}} 4 \times 5,~~ B {\it{~is~}} 4 \times 5,~~ C {\it{~is~}} 5 \times 2,~~ D {\it{~is~}} 4 \times 2,$

Determine whether the following matrix expressions are deﬁned, and, for those that are deﬁned, determine the size of the resulting matrix.

$(a)~ BA ~~~(b)~ AC + D ~~~(c)~ AE + B~~~ (d)~ AB + B~~~ (e)~E(A + B)~~~ (f) E(AC)$

4. Suppose that $$A, B, C, D,$$ and $$E$$ are the following matrices:

$A=\left[ \begin{array}{cc} 3 & 0 \\ -1 & 2 \\ 1&1 \end{array} \right],~ B= \left[ \begin{array}{cc} 4 & -1 \\ 0 & 2 \end{array} \right], ~ C= \left[ \begin{array}{ccc} 1 & 4 &2 \\ 3 & 1&5 \end{array} \right],\\ D= \left[ \begin{array}{ccc} 1 & 5 &2 \\ -1 & 0 & 1 \\ 3& 2 & 4\end{array} \right], {\it{~and ~}}E= \left[\begin{array}{ccc} 6 & 1 &3 \\ -1 & 1 & 2 \\ 4& 1 & 3\end{array} \right].$

Determine whether the following matrix expressions are deﬁned, and, for those that are deﬁned, compute the resulting matrix.

$$(a)~ D + E~~ (b)~ D - E~~ (c)~ 5A~~ (d)~ -7C~~ (e)~ 2B - C\\ (f)~ 2E - 2D~~ (g)~ -3(D + 2E)~~ (h)~A - A~~ (i)~ AB~~ (j)~ BA\\ (k)~ (3E)D~~ (l)~ (AB)C ~~(m)~ A(BC)~~ (n)~(4B)C + 2B ~~(o)~ D - 3E\\ (p)~ CA + 2E ~~(q)~ 4E - D ~~(r)~ DD$$

5. Suppose that $$A, B,$$ and $$C$$ are the following matrices and that $$a = 4$$ and $$b = 7.$$

$A= \left[ \begin{array}{ccc} 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 \end{array} \right],B = \left[ \begin{array}{ccc} 6 & 1 & 3 \\ -1 & 1 & 2 \\ 4 & 1 & 3 \end{array} \right], {\it{~and~}} C = \left[ \begin{array}{ccc} 1 & 5 & 2 \\ -1 & 0 & 1 \\ 3 & 2 & 4 \end{array} \right].$

Verify computationally that
$$(a)~ A + (B + C) = (A + B) + C ~~~(b) ~(AB)C = A(BC)\\ (c)~ (a + b)C = aC + bC ~~~(d)~ a(B - C) = aB - aC\\ (e)~ a(BC) = (aB)C = B(aC) ~~~(f)A(B - C) = AB - AC\\ (g)~ (B + C)A = BA + CA ~~~(h) a(bC) = (ab)C\\ (i)~ B - C = -C + B$$

6. Suppose that $$A$$ is the matrix
$A=\left[ \begin{array}{cc} 3 & 1 \\ 2 & 1 \end{array} \right]$
Compute $$p(A)$$, where $$p(z)$$ is given by
$$(a)~ p(z) = z - 2 ~~~(b)~ p(z) = 2z^2 - z + 1\\ (c)~ p(z) = z^3 - 2z + 4~~~ (d)~ p(z) = z^2 - 4z + 1$$

7. Deﬁne matrices $$A, B, C, D,$$ and $$E$$ by

$A=\left[ \begin{array}{cc} 3 & 1 \\ 2 & 1 \end{array} \right],~ B= \left[ \begin{array}{cc} 4 & -1 \\ 0 & 2 \end{array} \right], ~ C= \left[ \begin{array}{ccc} 2 & -3 &5 \\ 9 & -1&1 \\ 1&5&4\end{array} \right],\\ D= \left[ \begin{array}{ccc} 1 & 5 &2 \\ -1 & 0 & 1 \\ 3& 2 & 4\end{array} \right], {\it{~and ~}}E= \left[\begin{array}{ccc} 6 & 1 &3 \\ -1 & 1 & 2 \\ 4& 1 & 3\end{array} \right].$

(a) Factor each matrix into a product of elementary matrices and an RREF matrix.
(b) Find, if possible, the LU-factorization of each matrix.
(c) Determine whether or not each of these matrices is invertible, and, if possible, compute the inverse.

8. Suppose that $$A, B, C, D,$$ and $$E$$ are the following matrices:

$A=\left[ \begin{array}{cc} 3 & 0 \\ -1 & 2 \\ 1&1 \end{array} \right],~ B= \left[ \begin{array}{cc} 4 & -1 \\ 0 & 2 \end{array} \right], ~ C= \left[ \begin{array}{ccc} 1 & 4 &2 \\ 3 & 1&5 \end{array} \right],\\ D= \left[ \begin{array}{ccc} 1 & 5 &2 \\ -1 & 0 & 1 \\ 3& 2 & 4\end{array} \right], {\it{~and ~}}E= \left[\begin{array}{ccc} 6 & 1 &3 \\ -1 & 1 & 2 \\ 4& 1 & 3\end{array} \right].$

Determine whether the following matrix expressions are deﬁned, and, for those that are deﬁned, compute the resulting matrix.

$$(a)~ 2A^T + C~~~ (b)~ D^T - E^T~~~ (c)~ (D - E)^T\\ (d)~ B^T + 5C^T~~~ (e) ~\frac{1}{2}C^T - \frac{1}{4}A~~~ (f)~ B B^T\\ (g) ~3E^T - 3D^T~~~ (h)~ (2E^T - 3D^T )^T~~~ (i)~ CC^T\\ (j)~ (DA)^T~~~ (k)~ (C^TB)A^T~~~ (l)~ (2D^T - E)A\\ (m)~ (BA^T - 2C)^T~~~ (n)~ B^T (CC^T - A^TA)~~~ (o)~D^TE^T - (ED)^T\\ (p)~ trace(DD^T)~~~ (q)~trace(4E^T - D)~~~ (r)~trace(C^TA^T + 2E^T )$$

## Proof-Writing Exercises

1. Let $$n \in \mathbb{Z}_+$$ be a positive integer and $$a_{i,j} \in \mathbb{F}$$ be scalars for $$i, j = 1, \ldots , n.$$ Prove that
the following two statements are equivalent:
(a) The trivial solution $$x_1 = \cdots = x_n = 0$$ is the only solution to the homogeneous system of equations
$\left. \begin{array}{ccc} \sum_{k=1}^{n} a_{1,k}x_k & = & 0 \\ & \vdots & \\ \sum_{k=1}^{n} a_{n,k}x_k & = & 0 \end{array} \right\}.$

(b) For every choice of scalars $$c_1 , \ldots , c_n \in \mathbb{F},$$ there is a solution to the system of equations $\left. \begin{array}{ccc} \sum_{k=1}^{n} a_{1,k}x_k & = & c_1 \\ & \vdots & \\ \sum_{k=1}^{n} a_{n,k}x_k & = & c_n \end{array} \right\}.$
2. Let $$A$$ and $$B$$ be any matrices.
(a) Prove that if both $$AB$$ and $$BA$$ are deﬁned, then $$AB$$ and $$BA$$ are both square matrices.
(b) Prove that if $$A$$ has size $$m \times n$$ and $$ABA$$ is deﬁned, then $$B$$ has size $$n \times m.$$
3. Suppose that $$A$$ is a matrix satisfying $$A^T A = A.$$ Prove that $$A$$ is then a symmetric matrix and that $$A = A^2 .$$
4. Suppose $$A$$ is an upper triangular matrix and that $$p(z)$$ is any polynomial. Prove or give a counterexample: $$p(A)$$ is a upper triangular matrix.

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