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Chapter 6: Problem 59

Describe how to multiply matrices.

Short Answer

Expert verified
To multiply matrices, ensure the number of columns of the first matrix equals the number of rows of the second. Setup a resulting matrix with dimensions based on the first matrix's rows and the second matrix's columns. Each entry is the sum of the product of corresponding entries from the respective row of the first matrix and column of the second.

Step by step solution

01

Verify Matrix Dimensions

In order to multiply two matrices, the number of columns in the first matrix (matrix A) must equal the number of rows in the second matrix (matrix B). If matrix A is of size (m x n) and matrix B is of size (p x q), this condition is satisfied when n is equal to p. If this is not achieved, matrix multiplication cannot occur.
02

Setting Up the Resultant Matrix

Set-up an empty resultant matrix (matrix C) which has the same number of rows as matrix A and the same number of columns as matrix B. The size of matrix C would then be (m x q).
03

Calculate the Entries of Resultant Matrix

Each element in the resultant matrix must be calculated individually. The element in the i-th row and j-th column of the resultant matrix C is calculated as the sum of the product of corresponding elements from the i-th row of matrix A and the j-th column of matrix B. This process is repeated until all entries in matrix C are filled.

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Most popular questions from this chapter

In Exercises \(27-36,\) find (if possible): \(\begin{array}{llll}\text { a. } A B & \text { and } & \text { b. } B A\end{array}\) $$ A=\left[\begin{array}{llll} 1 & 2 & 3 & 4 \end{array}\right], \quad B=\left[\begin{array}{l} 1 \\ 2 \\ 3 \\ 4 \end{array}\right] $$

In Exercises \(17-26,\) let $$ A=\left[\begin{array}{rr} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{array}\right] \text { and } B=\left[\begin{array}{rr} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{array}\right] $$ Solve each matrix equation for \(X\). $$ 3 X+2 A=B $$

Evaluate each determinant. $$ \left|\begin{array}{rrr}1 & 1 & 1 \\\2 & 2 & 2 \\\\-3 & 4 & -5\end{array}\right| $$

Evaluate each determinant. $$ \left|\begin{array}{rrrr}3 & -1 & 1 & 2 \\\\-2 & 0 & 0 & 0 \\\2 & -1 & -2 & 3 \\\1 & 4 & 2 & 3\end{array}\right| $$

Use a graphing utility to evaluate the determinant for the given matrix. $$ \left[\begin{array}{rrrr}3 & -2 & -1 & 4 \\\\-5 & 1 & 2 & 7 \\\2 & 4 & 5 & 0 \\\\-1 & 3 & -6 & 5\end{array}\right] $$
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