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Chapter 8: Problem 83

Let \(A\) and \(B\) be unequal diagonal matrices of the same order. (A diagonal matrix is a square matrix in which each entry not on the main diagonal is zero.) Determine the products \(A B\) for several pairs of such matrices. Make a conjecture about a quick rule for such products.

Short Answer

Expert verified
The product of two unequal diagonal matrices A and B of the same order is another diagonal matrix of the same order. The elements on the diagonal of the product matrix are the product of the corresponding elements on the diagonals in A and B.

Step by step solution

01

Understand diagonal matrices

A diagonal matrix is a square matrix where all elements not in the main diagonal are zeros and the main diagonal can consist of either zeros or non-zeros. A diagonal matrix A of order n can be represented by \(A = [a_{ij}]\) where \(a_{ij} = 0\) for \(i \neq j\). An example of a 3x3 diagonal matrix: \[A = \begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix}\]
02

Multiply Two Diagonal matrices

Let two 3x3 diagonal matrices be A and B. \[A = \begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix}\] and \[B = \begin{bmatrix} d & 0 & 0 \ 0 & e & 0 \ 0 & 0 & f \end{bmatrix}\]. The product AB would be computed as \[AB = \begin{bmatrix} a*d & 0 & 0 \ 0 & b*e & 0 \ 0 & 0 & c*f \end{bmatrix}\] . It's clear that the product of two diagonal matrices of the same order is another diagonal matrix whose elements are the product of corresponding elements in the original matrices.
03

Conjecture a rule

Observing the product matrix, a rule can be conjectured: 'The product of two unequal diagonal matrices of the same order is another diagonal matrix of the same order. The elements of the product matrix are the product of the corresponding elements of the multiplying matrices.' This rule can be formulated mathematically as: if \(A = diag(a_1, a_2, ..., a_n)\) and \(B = diag(b_1, b_2, ..., b_n)\), then \(AB = diag(a_1*b_1, a_2*b_2, ..., a_n*b_n)\)

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Most popular questions from this chapter

Using a Graphing Utility, use the matrix capabilities of a graphing utility to write the augmented matrix corresponding to the system of equations in reduced row-echelon form. Then solve the system. $$\left\\{\begin{aligned} x+y+z+w &=0 \\ 2 x+3 y+z-2 w &=0 \\ 3 x+5 y+z &=0 \end{aligned}\right.$$

Conjecture A diagonal matrix is a square matrix with all zero entries above and below its main diagonal. Evaluate the determinant of each diagonal matrix. Make a conjecture based on your results. $$(\mathrm{a})\left[ \begin{array}{ll}{7} & {0} \\ {0} & {4}\end{array}\right] \quad(\mathrm{b}) \left[ \begin{array}{rrr}{-1} & {0} & {0} \\ {0} & {5} & {0} \\ {0} & {0} & {2}\end{array}\right] (\mathrm{c}) \left[ \begin{array}{rrrr}{2} & {0} & {0} & {0} \\ {0} & {-2} & {0} & {0} \\\ {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {3}\end{array}\right]$$

Comparing Linear Systems and Matrix Operations In Exercises 41 and \(42,\) (a) perform the row operations to solve the augmented matrix, (b) write and solve the system of linear equations represented by the augmented matrix, and (c) compare the two solution methods. Which do you prefer? $$\left[ \begin{array}{rrrr}{7} & {13} & {1} & {\vdots} & {-4} \\ {-3} & {-5} & {-1} & {\vdots} & {-4} \\ {3} & {6} & {1} & {\vdots} & {-2}\end{array}\right]$$ $$\begin{array}{l}{\text { (i) Add } R_{2} \text { to } R_{1} \text { . }} \\\ {\text { (ii) Multiply } R_{1} \text { by } \frac{1}{4}} \\ {\text { (iii) } \text { Add } R_{3} \text { to } R_{2} \text { . }}\end{array}$$ $$\begin{array}{l}{\text { (iv) } \mathrm{Add}-3 \text { times } R_{1} \text { to } R_{3} \text { . }} \\ {\text { (v) } \mathrm{Add}-2 \text { times } R_{2} \text { to } R_{1} .}\end{array}$$

Gaussian Elimination with Back-Substitution, use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution. $$\left\\{\begin{array}{rr}{x+2 y-3 z=} & {-28} \\ {4 y+2 z=} & {0} \\\ {-x+y-z=} & {-5}\end{array}\right.$$

HOW DO YOU SEEIT? Determine whether the matrix below is in row-echelon form, reduced row-echelon form, or neither when it satisfies the given conditions. $$\left[ \begin{array}{ll}{1} & {b} \\ {c} & {1}\end{array}\right]$$ $$\begin{array}{ll}{\text { (a) } b=0, c=0} & {\text { (b) } b \neq 0, c=0} \\\ {\text { (c) } b=0, c \neq 0} & {\text { (d) } b \neq 0, c \neq 0}\end{array}$$
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