Given A=[4−231],B=[510−237]A = \left[ \begin{matrix} 4 & -2 \\ 3 & 1 \end{matrix} \right], \quad B = \left[ \begin{matrix} 5 & 1 \\ 0 & -2 \\ 3 & 7 \end{matrix} \right] , and C=[−541036]C = \left[ \begin{matrix} -5 & 4 & 1 \\ 0 & 3 & 6 \end{matrix} \right] C=[−504316], find each product, if possible. See Examples 5–7. AB

Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

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Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 73

Chapter 6, Problem 73

Given $A = [\begin{matrix} 4 & - 2 \\ 3 & 1 \end{matrix}],$ , and $C = $\left$[ $\begin{matrix}$ -5 & 4 & 1 \\ 0 & 3 & 6 $\end{matrix}$ $\right$]$ , find each product, if possible. See Examples 5–7. AB

Verified step by step guidance

Identify the dimensions of matrices A and B. For matrix multiplication AB to be defined, the number of columns in A must equal the number of rows in B.

Write down the dimensions of A and B explicitly. For example, if A is an m×n matrix and B is a p×q matrix, then n must equal p for AB to be possible.

If the multiplication is possible, set up the product matrix AB, which will have dimensions m×q.

Calculate each element of the product matrix AB by taking the dot product of the corresponding row of A with the corresponding column of B. Specifically, the element in row i and column j of AB is given by $\sum_{k=1}^n A_{ik} \times B_{kj}$.

Perform the multiplication for each element systematically to fill the entire product matrix AB.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Matrix Dimensions and Compatibility

Matrix multiplication is only defined when the number of columns in the first matrix equals the number of rows in the second matrix. For example, if A is an m×n matrix and B is a p×q matrix, the product AB exists only if n = p.

Matrix Multiplication Procedure

To multiply two matrices, each element of the resulting matrix is computed as the dot product of the corresponding row from the first matrix and the column from the second matrix. This involves multiplying corresponding entries and summing the results.

Resulting Matrix Dimensions

The product of an m×n matrix and an n×p matrix results in an m×p matrix. Understanding the size of the resulting matrix helps in verifying the correctness of the multiplication and organizing the computation.

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Related Practice

Textbook Question

Solve each system. (Hint: In Exercises 69–72, let $1 / x = t$ and $1 / y = u$ .)

$\frac{2}{x} + \frac{3}{y} = 18$

$\frac{4}{x} - \frac{5}{y} = - 8$

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Textbook Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

(1/2)x + (1/3)y = 2

(3/2)x - (1/2)y = -12

Textbook Question

Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = √(0.1q + 9) - 2 and demand: p = √(25 - 0.1q).

Find the equilibrium demand.

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Textbook Question

Perform each operation, if possible.

$[\begin{matrix} 3 \\ 2 \\ 5 \end{matrix}] - [\begin{matrix} 8 \\ - 4 \\ 6 \end{matrix}] + [\begin{matrix} 1 \\ 0 \\ 2 \end{matrix}]$

Textbook Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

2x - y + 4z = -2

3x + 2y - z = -3

x + 4y - 2z = 17

Textbook Question

Solve each problem. The supply and demand equations for a certain commodity are given. supply: p = √(0.1q + 9) - 2 and demand: p = √(25 - 0.1q).

Find the equilibrium price (in dollars).