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. $$ \begin{array}{l} A=\left[\begin{array}{ccc} -4 & 3 & 3 \\ -5 & -1 & -5 \\ -5 & 0 & -1 \end{array}\right] \\ B=\left[\begin{array}{ccc} 0 & 5 & 0 \\ -5 & -4 & 3 \\ 5 & -4 & 3 \end{array}\right] \end{array} $$

Short answer

Both AB and BA are possible and each is a 3x3 matrix. AB = [[0, -53, 18], [-20, -21, -18], [-5, -21, -3]], BA = [[-25, -5, -25], [61, -11, 23], [15, 19, 63]].

Step-by-step solution

Determine the Dimensions of Matrix A

Matrix \( A \) is given as a \( 3 \times 3 \) matrix because it has 3 rows and 3 columns.

Determine the Dimensions of Matrix B

Similarly, matrix \( B \) is also a \( 3 \times 3 \) matrix with 3 rows and 3 columns.

Check the Possibility of Matrix Multiplication

For the product \( A B \) to be defined, the number of columns in \( A \) (which is 3) must equal the number of rows in \( B \) (which is also 3). Thus, \( A B \) is possible and will be a \( 3 \times 3 \) matrix.For the product \( B A \) to be defined, the number of columns in \( B \) must equal the number of rows in \( A \). This is also true, so \( B A \) will also be a \( 3 \times 3 \) matrix.

Compute the Product AB

To compute \( A B \), we multiply each row of \( A \) by each column of \( B \), summing the products. Calculating each element:- First row, first column: \(-4 \times 0 + 3 \times (-5) + 3 \times 5 = 0\).- First row, second column: \(-4 \times 5 + 3 \times (-4) + 3 \times (-4) = -53\).- First row, third column: \(-4 \times 0 + 3 \times 3 + 3 \times 3 = 18\).Calculating the rest similarly: - Second row, first column: \(-5 \times 0 + (-1) \times (-5) + (-5) \times 5 = -20\).- Second row, second column: \(-5 \times 5 + (-1) \times (-4) + (-5) \times (-4) = -21\).- Second row, third column: \(-5 \times 0 + (-1) \times 3 + (-5) \times 3 = -18\).- Third row, first column: \(-5 \times 0 + 0 \times (-5) + (-1) \times 5 = -5\).- Third row, second column: \(-5 \times 5 + 0 \times (-4) + (-1) \times (-4) = -21\).- Third row, third column: \(-5 \times 0 + 0 \times 3 + (-1) \times 3 = -3\).So the product \( A B \) is:\[\begin{bmatrix}0 & -53 & 18 \-20 & -21 & -18 \-5 & -21 & -3 \\end{bmatrix}\]

Compute the Product BA

To compute \( B A \), we multiply each row of \( B \) by each column of \( A \), summing the results.Calculating each element:- First row, first column: \(0 \times (-4) + 5 \times (-5) + 0 \times (-5) = -25\).- First row, second column: \(0 \times 3 + 5 \times (-1) + 0 \times 0 = -5\).- First row, third column: \(0 \times 3 + 5 \times (-5) + 0 \times (-1) = -25\).Calculating the rest similarly:- Second row, first column: \(-5 \times (-4) + (-4) \times (-5) + 3 \times (-5) = 61\).- Second row, second column: \(-5 \times 3 + (-4) \times (-1) + 3 \times 0 = -11\).- Second row, third column: \(-5 \times 3 + (-4) \times (-5) + 3 \times (-1) = 23\).- Third row, first column: \(5 \times (-4) + (-4) \times (-5) + 3 \times (-5) = 15\).- Third row, second column: \(5 \times 3 + (-4) \times (-1) + 3 \times 0 = 19\).- Third row, third column: \(5 \times 3 + (-4) \times (-5) + 3 \times (-1) = 63\).So the product \( B A \) is:\[\begin{bmatrix}-25 & -5 & -25 \61 & -11 & 23 \15 & 19 & 63 \\end{bmatrix}\]

Key concepts

Matrix Dimensions

Understanding matrix dimensions is a foundational concept in matrix multiplication. Each matrix is defined by its dimensions, which tell us how many rows and columns it has. For example, if a matrix is described as a "3x3 matrix," this means it contains 3 rows and 3 columns. In the context of the exercise above, both matrices \( A \) and \( B \) are 3x3 matrices.

Recognizing the dimensions is essential because it helps us identify whether two matrices can be multiplied. The rule for matrix multiplication requires the number of columns in the first matrix to be equal to the number of rows in the second. Given our matrices are both 3x3, this condition is satisfied for both the multiplication of \( AB \) and \( BA \). Therefore, it becomes possible to compute both products.

In summary, always start by checking and understanding the dimensions of your matrices as this determines the feasibility of performing the multiplication operation.

Matrix Products

Matrix products involve calculating a new matrix from two given matrices. To form a matrix product, such as \( AB \), we multiply each element of the rows of the first matrix by each corresponding element of the columns of the second matrix and sum the results.

This is done for every row-column pair to form the new matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. In our case, \( AB \) and \( BA \) both result in new 3x3 matrices.

• For \( AB\), you take a row from \( A\) and a column from \( B\). Multiply corresponding elements and sum them up to get the new matrix element.
• Perform the same operation for all row-column combinations to complete the matrix.

For example, the element in the first row and first column of \( AB \) is calculated as \(-4 \times 0 + 3 \times (-5) + 3 \times 5\), summing up to zero. This process is repeated for each element in the resulting matrix.

Practicing this technique will strengthen understanding of how matrix multiplication works, leading to ease in handling larger matrices and more complex problems.

Matrix Operations

Matrix operations, such as addition, subtraction, and multiplication, expand your ability to work with data stored in matrices. The exercise presented revolves around matrix multiplication, which is crucial in various mathematical and real-world applications, including computer graphics and solving systems of equations.

Multiplying matrices isn't the only operation, but it's one of the most crucial and complex matrix operations. It combines information from two different matrices into a single matrix which often represents a new data set.

• Addition: Matrices can be added together if they are of the same dimensions. Add corresponding elements from each matrix.
• Subtraction: Like addition, subtract corresponding elements from matrices of the same dimensions.
• Multiplication: Can only be done when the number of columns in the first matrix matches the number of rows in the second.

When performing these operations, maintaining accuracy is crucial. Each miscalculation can significantly affect the outcome of your result matrix, especially with more complex calculations involved in multiplication.

Grasping these basic operations thoroughly can help set a solid foundation for more advanced topics in linear algebra and support applications within other areas of mathematics and engineering.