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

Hello, everyone. We are asked to perform the multiplication of the matrices A B if matrix A matrix B and matrix C are defined as follows. Matrix A is defined as a two by two matrix with the top row being negative six negative five. And the second row being 72, matrix B is a three by two matrix where the first row is 85, the second row is zero negative four. The third row is 19. As we are looking to multiply A and B, we do not need to worry about what matrix C says. We have four answer choices three of which are three by two matrices with slightly varying answers in them. And one answer choice D says this is not possible. So let's find out which one it is first recall that when we multiply matrices, the columns of the first matrix must match the rows of the second matrix. So A is a two by two. Matrix B is a three by two matrix two and three do not match. And such this is actually not possible because their dimensions do not match to make them be able to be multiplied So answer choice. D not possible is our answer. Have a nice day.

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

College Algebra

7. Systems of Equations & Matrices

Determinants and Cramer's Rule

1:48 minutes

Problem 73

Textbook Question

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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