Chapter 5

For the given matrices \(A\) find \(A^{-1}\) if it exists and verify that \(A A^{-1}=\) \(A^{-1} A=I .\) If \(A^{-1}\) does not exist explain why. (a) \(A=\left(\begin{array}{ll}1 & 3 \\ 2 & 1\end{array}\right)\) (b) \(A=\left(\begin{array}{ll}6 & -3 \\ 8 & -4\end{array}\right)\) (c) \(A=\left(\begin{array}{cc}1 & -3 \\ 0 & 1\end{array}\right)\) (d) \(A=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\) (e) Use the definition of the inverse of a matrix to find \(A^{-1}: A=\) $$ \left(\begin{array}{ccc} 3 & 0 & 0 \\ 0 & \frac{1}{2} & 0 \\ 0 & 0 & -5 \end{array}\right) $$

Let \(A=\left(\begin{array}{ccc}1 & 0 & 2 \\ 2 & -1 & 5 \\ 3 & 2 & 1\end{array}\right), B=\left(\begin{array}{ccc}0 & 2 & 3 \\ 1 & 1 & 2 \\ -1 & 3 & -2\end{array}\right),\) and \(C=\left(\begin{array}{cccc}2 & 1 & 2 & 3 \\\ 4 & 0 & 1 & 1 \\ 3 & -1 & 4 & 1\end{array}\right)\) Compute, if possible; (a) \(A-B\) (e) \(C A-C B\) (b) \(A B\) (c) \(A C-B C\) (f) \(C\left(\begin{array}{c}x \\ y \\ z \\ w\end{array}\right)\) (d) \(A(B C)\)

For the given matrices \(A\) find \(A^{-1}\) if it exists and verify that \(A A^{-1}=\) \(A^{-1} A=I\). If \(A^{-1}\) does not exist explain why. (a) \(A=\left(\begin{array}{cc}2 & -1 \\ -1 & 2\end{array}\right)\) (b) \(A=\left(\begin{array}{ll}0 & 1 \\ 0 & 2\end{array}\right)\) (c) \(A=\left(\begin{array}{ll}1 & c \\ 0 & 1\end{array}\right)\) (d) \(A=\left(\begin{array}{cc}a & b \\ b & a\end{array}\right),\) where \(|a| \neq|b|\).

Let \(A=\left(\begin{array}{cc}2 & 0 \\ 0 & 3\end{array}\right) .\) Find a matrix \(B\) such that \(A B=I\) and \(B A=I,\) where \(I=\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right)\)

Let \(A=\left(\begin{array}{cc}1 & 2 \\ 0 & -1\end{array}\right), B=\left(\begin{array}{ccc}3 & 7 & 6 \\ 2 & -1 & 5\end{array}\right),\) and \(C=\left(\begin{array}{ccc}0 & -2 & 4 \\ 7 & 1 & 1\end{array}\right) .\) Compute the following as efficiently as possible by using any of the Laws of Matrix Algebra: (a) \(A B+A C\) (b) \(A^{-1}\) (c) \(A(B+C)\) (d) \(\left(A^{2}\right)^{-1}\) (e) \((C+B)^{-1} A^{-1}\)

Prove or disprove the following implications. (a) \(A^{2}=A\) and \(\operatorname{det} A \neq 0 \Rightarrow A=I\) (b) \(A^{2}=I\) and \(\operatorname{det} A \neq 0 \Rightarrow A=I\) or \(A=-I\).

Let \(A=\left(\begin{array}{ll}7 & 4 \\ 2 & 1\end{array}\right)\) and \(B=\left(\begin{array}{ll}3 & 5 \\ 2 & 4\end{array}\right)\). Compute the following as effi- ciently as possible by using any of the Laws of Matrix Algebra: (a) \(A B\) (b) \(A+B\) (c) \(A^{2}+A B+B A+B^{2}\) (d) \(B^{-1} A^{-1}\) (e) \(A^{2}+A B\)

Let \(M_{n \times n}(\mathbb{R})\) be the set of real \(n \times n\) matrices. Let \(P \subseteq M_{n \times n}(\mathbb{R})\) be the subset of matrices defined by \(A \in P\) if and only if \(A^{2}=A .\) Let \(Q \subseteq P\) be defined by \(A \in Q\) if and only if \(\operatorname{det} A \neq 0\) (a) Determine the cardinality of \(Q\). (b) Consider the special case \(n=2\) and prove that a sufficient condition for \(A \in P \subseteq M_{2 \times 2}(\mathbb{R})\) is that \(A\) has a zero determinant (i.e., \(A\) is singular) and \(\operatorname{tr}(A)=1\) where \(\operatorname{tr}(A)=a_{11}+a_{22}\) is the sum of the main diagonal elements of \(A\). (c) Is the condition of part b a necessary condition?

Let \(A=\left(\begin{array}{cc}a & b \\ c & d\end{array}\right) .\) Derive the formula for \(A^{-1}\).

Find \(A^{3}\) if \(A=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{array}\right)\). What is \(A^{15}\) equal to?

Let \(A\) and \(B\) be \(n \times n\) matrices of real numbers. Is \(A^{2}-B^{2}=(A-B)(A+\) \(B) ?\) Explain.

(a) Determine \(I^{2}\) and \(I^{3}\) if \(I=\left(\begin{array}{ccc}1 & 0 & 0 \\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)\). (b) What is \(I^{n}\) equal to for any \(n \geq 1 ?\) (c) Prove your answer to part (b) by induction.

Recall that \(p(x)=x^{2}-5 x+6\) is called a polynomial, or more specifically, a polynomial over \(\mathbb{R},\) where the coefficients are elements of \(\mathbb{R}\) and \(x \in \mathbb{R}\). Also, think of the method of solving, and solutions of, \(x^{2}-5 x+6=0\). We would like to define the analogous situation for \(2 \times 2\) matrices. First define where \(A\) is a \(2 \times 2\) matrix \(p(A)=A^{2}-5 A+6 I\). Discuss the method of solving and the solutions of \(A^{2}-5 A+6 I=0\).

Prove by induction that for \(n \geq 1,\left(\begin{array}{cc}a & 0 \\ 0 & b\end{array}\right)^{n}=\left(\begin{array}{cc}a^{n} & 0 \\ 0 & b^{n}\end{array}\right)\).

(a) If $$ A=\left(\begin{array}{cc} 2 & 1 \\ 1 & -1 \end{array}\right), X=\left(\begin{array}{l} x_{1} \\ x_{2} \end{array}\right), \text { and } B=\left(\begin{array}{l} 3 \\ 1 \end{array}\right) $$ \(2 x_{1}+x_{2}=3\) show that \(A X=B\) is a way of expressing the system \(x_{1}-x_{2}=1\) using matrices. (b) Express the following systems of equations using matrices: (i) \(2 x_{1}-x_{2}=4\) $$ x_{1}+x_{2} \quad=3 $$ $$ x_{1}+x_{2}=0 $$ (iii) \(x_{2} \quad=5\) $$ x_{1}+x_{2}+2 x_{3}=1 \quad x_{1} \quad+3 x_{3}=6 $$ (ii) \(x_{1}+2 x_{2}-x_{3}=-1\) \(x_{1}+3 x_{3}+x_{3}=5\)

In this exercise, we propose to show how matrix multiplication is a natural operation. Suppose a bakery produces bread, cakes and pies every weekday, Monday through Friday. Based on past sales history, the bakery produces various numbers of each product each day, summarized in the \(5 \times 3\) matrix \(D\). It should be noted that the order could be described as "number of days by number of products." For example, on Wednesday (the third day) the number of cakes (second product in our list) that are produced is \(d_{3,2}=4\). $$ D=\left(\begin{array}{ccc} 25 & 5 & 5 \\ 14 & 5 & 8 \\ 20 & 4 & 15 \\ 18 & 5 & 7 \\ 35 & 10 & 9 \end{array}\right) $$ The main ingredients of these products are flour, sugar and eggs. We assume that other ingredients are always in ample supply, but we need to be sure to have the three main ones available. For each of the three products, The amount of each ingredient that is needed is summarized in the \(3 \times 3\), or "number of products by number of ingredients" matrix \(P\). For example, to bake a cake (second product) we need \(P_{2,1}=1.5\) cups of flour (first ingredient). Regarding units: flour and sugar are given in cups per unit of each product, while eggs are given in individual eggs per unit of each product. $$ P=\left(\begin{array}{ccc} 2 & 0.5 & 0 \\ 1.5 & 1 & 2 \\ 1 & 1 & 1 \end{array}\right) $$ These amounts are "made up", so don't used them to do your own baking! (a) How many cups of flour will the bakery need every Monday? Pay close attention to how you compute your answer and the units of each number. (b) How many eggs will the bakery need every Wednesday? (c) Compute the matrix product \(D P\). What do you notice? (d) Suppose the costs of ingredients are \(\$ 0.12\) for a cup of flour, \(\$ 0.15\) for a cup of sugar and \(\$ 0.19\) for one egg. How can this information be put into a matrix that can meaningfully be multiplied by one of the other matrices in this problem?

Prove: If the determinant of a matrix \(A\) is zero, then \(A\) does not have an inverse. Hint: Use the indirect method of proof and exercise 5 .

(a) Let \(A, B,\) and \(D\) be \(n \times n\) matrices. Assume that \(B\) is invertible. If \(A=B D B^{-1},\) prove by induction that \(A^{m}=B D^{m} B^{-1}\) is true for \(m \geq 1\) (b) Given that \(A=\left(\begin{array}{ll}-8 & 15 \\ -6 & 11\end{array}\right)=B\left(\begin{array}{ll}1 & 0 \\ 0 & 2\end{array}\right) B^{-1}\) where \(B=\) \(\left(\begin{array}{ll}5 & 3 \\ 3 & 2\end{array}\right)\) what is \(A^{10} ?\)