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Chapter 2: Problem 7

If a matrix \(A\) is \(5 \times 3\) and the product \(A B\) is \(5 \times 7,\) what is the size of \(B ?\)

Short Answer

Expert verified
Matrix B is 3 x 7.

Step by step solution

01

Understand Matrix Multiplication

For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. If matrix A is of size \(m \times n\) and the resultant product matrix \(AB\) is of size \(m \times p\), then matrix B must be of size \(n \times p\).
02

Determine the Dimensions of A

Matrix \(A\) is given to be of dimension \(5 \times 3\). This means \(A\) has 5 rows and 3 columns.
03

Analyze the Resultant Matrix Dimensions

The product matrix \(A B\) is of size \(5 \times 7\). This indicates that the resulting matrix has 5 rows and 7 columns.
04

Determine the Required Dimensions of B

Based on the rules of matrix multiplication and the provided dimensions, matrix \(B\) must have 3 rows to match the number of columns in \(A\) and 7 columns to match the number of columns in the product \(A B\). Therefore, \(B\) must be \(3 \times 7\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Dimensions
In matrix mathematics, dimensions are crucial to understanding how matrices interact, especially when it comes to operations like multiplication. A matrix's dimensions are determined by counting its rows and columns. Often, dimensions are described in the format of "m x n," where "m" is the number of rows, and "n" is the number of columns.
Matrix dimensions help us quickly comprehend the structure and size of the matrix. For instance, a matrix with dimensions 5 x 3 has 5 horizontal rows and 3 vertical columns. This convention is very important when multiplying matrices, as it directly influences what matrices can be multiplied together.
Dimension Analysis
When analyzing the dimensions of matrices for multiplication, a specific condition must be met: the number of columns in the first matrix must equal the number of rows in the second matrix. This rule ensures that every element in a row of the first matrix can multiply with every corresponding element in a column of the second matrix.
To perform dimension analysis effectively:
  • Check the dimensions of the first matrix, say matrix A, denoted as "m x n".
  • Confirm that the second matrix, matrix B, must have dimensions "n x p" where "n" corresponds to matrix A鈥檚 columns.
  • This ensures each element aligns correctly for multiplication.
If the dimensions do not align following these rules, matrix multiplication is not possible.
Resultant Matrix Size
After successful multiplication of matrices, the resulting matrix has its own dimensions. These dimensions are determined by the outer numbers of the considering matrices. In our example, matrix A is 5 x 3 and matrix B must be 3 x 7 for multiplication to occur, here "3" is common ensuring multiplication rules are met.
The resultant matrix, denoted as AB, then takes the dimensions from the number of rows of A and the number of columns of B. Hence, the resultant size is 5 x 7.
Understanding resultant matrix size is crucial for anticipating the output of matrix multiplication operations. It helps to visualize and plan for the space required for the resulting data.

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Most popular questions from this chapter

In Exercises 17 and \(18,\) mark each statement True or False. Justify each answer. Here \(A\) is an \(m \times n\) matrix. a. If \(\mathcal{B}\) is a basis for a subspace \(H,\) then each vector in \(H\) can be written in only one way as a linear combination of the vectors in \(\mathcal{B}\) . bectors in \(\mathcal{B}\) . b. If \(\mathcal{B}=\left\\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\\}\) is a basis for a subspace \(H\) of \(\mathbb{R}^{n},\) then the correspondence \(\mathbf{x} \mapsto[\mathbf{x}]_{\mathcal{B}}\) makes \(H\) look and act the same as \(\mathbb{R}^{p}\) . c. The dimension of Nul \(A\) is the number of variables in the equation \(A \mathbf{x}=\mathbf{0} .\) d. The dimension of the column space of \(A\) is rank \(A .\) e. If \(H\) is a \(p\) -dimensional subspace of \(\mathbb{R}^{n},\) then a linearly independent set of \(p\) vectors in \(H\) is a basis for \(H .\)

In Exercises 19鈥�24, justify each answer or construction. If possible, construct a \(3 \times 4\) matrix \(A\) such that dim Nul \(A=2\) and \(\operatorname{dim} \operatorname{Col} A=2\)

In Exercises 37 and 38, construct bases for the column space and the null space of the given matrix A. Justify your work. $$ A=\left[\begin{array}{rrrrr}{5} & {2} & {0} & {-8} & {-8} \\ {4} & {1} & {2} & {-8} & {-9} \\ {5} & {1} & {3} & {5} & {19} \\ {-8} & {-5} & {6} & {8} & {5}\end{array}\right] $$

In Exercises 11 and \(12,\) give integers \(p\) and \(q\) such that Nul \(A\) is a subspace of \(\mathbb{R}^{p}\) and \(\mathrm{Col} A\) is a subspace of \(\mathbb{R}^{q}\) . $$ A=\left[\begin{array}{rrr}{1} & {2} & {3} \\ {4} & {5} & {7} \\ {-5} & {-1} & {0} \\ {2} & {7} & {11}\end{array}\right] $$

In Exercises \(3-8,\) find the \(3 \times 3\) matrices that produce the described composite 2 \(\mathrm{D}\) transformations, using homogeneous coordinates. Translate by \((3,1),\) and then rotate \(45^{\circ}\) about the origin.
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