Question

Matrices \(A\) and \(B\) are defined. (a) Give the dimensions of \(A\) and \(B\). If the dimensions properly match, give the dimensions of \(A B\) and \(B A\). (b) Find the products \(A B\) and \(B A\), if possible. $$ \begin{array}{l} A=\left[\begin{array}{ccc} -1 & 2 & 1 \\ -1 & 2 & -1 \\ 0 & 0 & -2 \end{array}\right] \\ B=\left[\begin{array}{ccc} 0 & 0 & -2 \\ 1 & 2 & -1 \\ 1 & 0 & 0 \end{array}\right] \end{array} $$

Short answer

Dimensions of \(A\) and \(B\) are \(3 \times 3\). \(AB = \begin{bmatrix} 3 & 4 & 0 \\ 1 & 4 & 0 \\ -2 & 0 & 0 \end{bmatrix}\) and \(BA = \begin{bmatrix} 0 & 0 & 4 \\ -3 & 6 & 1 \\ -1 & 2 & 1 \end{bmatrix}\).

Step-by-step solution

Determine Dimensions of A and B

Matrix \(A\) has 3 rows and 3 columns, so its dimensions are \(3 \times 3\). Matrix \(B\) also has 3 rows and 3 columns, making its dimensions \(3 \times 3\).

Establish Conditions for Multiplication

Matrix multiplication is possible if the number of columns in the first matrix equals the number of rows in the second matrix. Since both \(A\) and \(B\) are \(3 \times 3\) matrices, we can compute \(AB\) and \(BA\).

Determine Dimensions of AB and BA

Since both \(A\) and \(B\) are \(3 \times 3\), the resultant matrices \(AB\) and \(BA\) will both have dimensions \(3 \times 3\).

Compute Product AB

To multiply \(A\) by \(B\), take the dot product of the rows of \(A\) with the columns of \(B\). Calculate each element of the resultant matrix:- First row, first column: \((-1)(0) + (2)(1) + (1)(1) = 0 + 2 + 1 = 3\)- First row, second column: \((-1)(0) + (2)(2) + (1)(0) = 0 + 4 + 0 = 4\)- First row, third column: \((-1)(-2) + (2)(-1) + (1)(0) = 2 - 2 + 0 = 0\)- Second row, first column: \((-1)(0) + (2)(1) + (-1)(1) = 0 + 2 - 1 = 1\)- Second row, second column: \((-1)(0) + (2)(2) + (-1)(0) = 0 + 4 + 0 = 4\)- Second row, third column: \((-1)(-2) + (2)(-1) + (-1)(0) = 2 - 2 + 0 = 0\)- Third row, first column: \((0)(0) + (0)(1) + (-2)(1) = 0 + 0 - 2 = -2\)- Third row, second column: \((0)(0) + (0)(2) + (-2)(0) = 0 + 0 + 0 = 0\)- Third row, third column: \((0)(-2) + (0)(-1) + (-2)(0) = 0 + 0 + 0 = 0\)So, \(AB = \begin{bmatrix} 3 & 4 & 0 \ 1 & 4 & 0 \ -2 & 0 & 0 \end{bmatrix}\).

Compute Product BA

To multiply \(B\) by \(A\), take the dot product of the rows of \(B\) with the columns of \(A\). Calculate each element of the resultant matrix:- First row, first column: \((0)(-1) + (0)(-1) + (-2)(0) = 0 + 0 + 0 = 0\)- First row, second column: \((0)(2) + (0)(2) + (-2)(0) = 0 + 0 + 0 = 0\)- First row, third column: \((0)(1) + (0)(-1) + (-2)(-2) = 0 + 0 + 4 = 4\)- Second row, first column: \((1)(-1) + (2)(-1) + (-1)(0) = -1 - 2 + 0 = -3\)- Second row, second column: \((1)(2) + (2)(2) + (-1)(0) = 2 + 4 + 0 = 6\)- Second row, third column: \((1)(1) + (2)(-1) + (-1)(-2) = 1 - 2 + 2 = 1\)- Third row, first column: \((1)(-1) + (0)(-1) + (0)(0) = -1 + 0 + 0 = -1\)- Third row, second column: \((1)(2) + (0)(2) + (0)(0) = 2 + 0 + 0 = 2\)- Third row, third column: \((1)(1) + (0)(-1) + (0)(-2) = 1 + 0 + 0 = 1\)So, \(BA = \begin{bmatrix} 0 & 0 & 4 \ -3 & 6 & 1 \ -1 & 2 & 1 \end{bmatrix}\).

Key concepts

Matrix Dimensions

Understanding the dimensions of a matrix is crucial for performing matrix operations. A matrix is a rectangular array of numbers organized into rows and columns. The dimensions of a matrix are defined as "rows" by "columns". For instance, if a matrix has 3 rows and 3 columns, it is called a 3x3 matrix.
In the given exercise, both matrices \(A\) and \(B\) are 3x3 matrices, meaning they each have 3 rows and 3 columns. The dimensions determine whether two matrices can be multiplied: the number of columns in the first matrix must match the number of rows in the second matrix. Here, since both matrices are 3x3, they can be multiplied in either order: \(AB\) or \(BA\). This results in both products being 3x3 matrices.

Dot Product

The dot product is a fundamental operation in matrix multiplication. It involves multiplying corresponding elements of two sequences and summing these products. In matrix multiplication, the dot product is used to determine each element of the resulting matrix.
When multiplying two matrices, the element at position \((i, j)\) in the resulting matrix is found by taking the dot product of the \(i^{th}\) row of the first matrix and the \(j^{th}\) column of the second matrix.

• For example, to find the first element of matrix \(AB\), you take the first row of matrix \(A\) and the first column of matrix \(B\), multiply the corresponding elements, and add the results: \((-1)(0) + (2)(1) + (1)(1) = 3\).

This operation is repeated for each position in the resulting matrix, making matrix multiplication computationally thorough but straightforward once you understand the dot product.

3x3 Matrices

3x3 matrices are a type of square matrix where the number of rows and columns is equal, specifically three in this case. They are commonly used in a variety of mathematical computations, including solving systems of linear equations, transformations in geometry, and computer graphics.
The structure of a 3x3 matrix allows for operations like addition, subtraction, and multiplication with other 3x3 matrices. In this exercise, both matrices \(A\) and \(B\) are 3x3, making their products \(AB\) and \(BA\) also 3x3 matrices.

• Multiplying 3x3 matrices involves quite a few computations, as each element in the resulting matrix depends on the dot product of corresponding rows and columns from the original matrices.

This uniformity in dimensions simplifies many matrix operations, but it's important to apply them correctly to achieve accurate results.

Matrix Operations

Matrix operations include a variety of calculations that can be performed on matrices, such as addition, subtraction, and multiplication. The specific rules, particularly for multiplication, require a keen understanding of matrix dimensions and the suitability of matrices for these operations.
When multiplying matrices, it's important that the number of columns in the first matrix matches the number of rows in the second. This condition allows for the execution of the dot product necessary at each step of creating the product matrix.

• The products \(AB\) and \(BA\) demonstrate how matrices can differently interact based on their position in the operation. Matrices are not commutative under multiplication (i.e., \(AB\) does not necessarily equal \(BA\)).

This exercise involving 3x3 matrices provides a practical example of performing matrix operations and understanding their outcomes.