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Chapter 2: Problem 6

Find condition(s) on the size of a matrix \(A\) such that \(A^{2}\) (that is, \(A A\) ) is defined.

Short Answer

Expert verified
In order for A虏 (A * A) to be defined, matrix A must be a square matrix, with the number of rows (m) equal to the number of columns (n). Therefore, the size of matrix A must be m脳m or n脳n (m = n).

Step by step solution

01

Understand matrix multiplication

In matrix multiplication, if we have a matrix A of size m脳n and a matrix B of size p脳q, the product AB will be defined if and only if n = p. If this condition is met, the resulting matrix of the product AB will have the size m脳q.
02

Apply the condition to A虏

Since we are looking at A虏 (A multiplied by A), both the first and the second matrices in the product are the same matrix A. Therefore, the dimensions of matrix A are m脳n, and we are looking to find the size of A such that A虏 is defined. For this to be true, based on the condition from Step 1, the number of columns in A (n) must equal the number of rows in A (m). So, we have: m = n
03

State the condition for A虏 to be defined

As per our analysis, a matrix A will satisfy the condition for A虏 (A * A) to be defined when the number of rows (m) is equal to the number of columns (n). In other words, matrix A must be a square matrix. That is, the size of matrix A must be m脳m or n脳n (m = n).

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Most popular questions from this chapter

The problems in exercise correspond to those in exercises 15-27, Section 2.1. Use the results of your previous work to help you solve these problems. The annual returns on Sid Carrington's three investments amounted to $$\$ 21,600$$: \(6 \%\) on a savings account, \(8 \%\) on mutual funds, and \(12 \%\) on bonds. The amount of Sid's investment in bonds was twice the amount of his investment in the savings account, and the interest earned from his investment in bonds was equal to the dividends he received from his investment in mutual funds. Find how much money he placed in each type of investment.

Let $$ \begin{array}{l} A=\left[\begin{array}{lll} 0 & 3 & 0 \\ 1 & 0 & 1 \\ 0 & 2 & 0 \end{array}\right] \quad B=\left[\begin{array}{rrr} 2 & 4 & 5 \\ 3 & -1 & -6 \\ 4 & 3 & 4 \end{array}\right] \\ C=\left[\begin{array}{rrr} 4 & 5 & 6 \\ 3 & -1 & -6 \\ 2 & 2 & 3 \end{array}\right] \end{array} $$ a. Compute \(A B\). b. Compute \(A C\). c. Using the results of parts (a) and (b), conclude that \(A B=A C\) does not imply that \(B=C\).

Solve the system of linear equations using the Gauss-Jordan elimination method. $$ \begin{array}{l} 2 x+3 y-2 z=10 \\ 3 x-2 y+2 z=0 \\ 4 x-y+3 z=-1 \end{array} $$

Write the given system of linear equations in matrix form. $$ \begin{array}{r} 2 x-3 y+4 z=6 \\ 2 y-3 z=7 \\ x-y+2 z=4 \end{array} $$

Write the given system of linear equations in matrix form. $$ \begin{array}{rr} x-2 y+3 z= & -1 \\ 3 x+4 y-2 z= & 1 \\ 2 x-3 y+7 z= & 6 \end{array} $$
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