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. $$ \begin{array}{l} A=\left[\begin{array}{cc} -1 & 5 \\ 6 & 7 \end{array}\right] \\ B=\left[\begin{array}{cccc} 5 & -3 & -4 & -4 \\ -2 & -5 & -5 & -1 \end{array}\right] \end{array} $$

Short answer

The dimensions of \( AB \) are \( 2 \times 4 \), and it is \( \begin{bmatrix} -15 & -22 & -21 & -1 \\ 16 & -53 & -59 & -31 \end{bmatrix} \). \( BA \) is not possible.

Step-by-step solution

Find Dimensions of Matrix A

Matrix \( A \) is given as \( \begin{bmatrix} -1 & 5 \ 6 & 7 \end{bmatrix} \). To find the dimensions, count the rows and columns. Matrix \( A \) has 2 rows and 2 columns. Thus, the dimensions of \( A \) are \( 2 \times 2 \).

Find Dimensions of Matrix B

Matrix \( B \) is given as \( \begin{bmatrix} 5 & -3 & -4 & -4 \ -2 & -5 & -5 & -1 \end{bmatrix} \). To find the dimensions, count the rows and columns. Matrix \( B \) has 2 rows and 4 columns. Thus, the dimensions of \( B \) are \( 2 \times 4 \).

Determine Possibility of AB and BA

For matrix multiplication to be possible between two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. For \( AB \) to be possible, the columns in \( A \) (2) must match the rows in \( B \) (2), which they do. So, \( AB \) is possible. \[ AB \text{ dimensions: } 2 \times 4 \]\ For \( BA \) to be possible, the columns in \( B \) (4) must match the rows in \( A \) (2), which they do not. So, \( BA \) is not possible.

Multiply A and B to Find AB

Since \( AB \) is possible, we calculate it by multiplying corresponding elements and summing them:- The element in the 1st row, 1st column: \((-1)(5) + (5)(-2) = -5 - 10 = -15\).- The element in the 1st row, 2nd column: \((-1)(-3) + (5)(-5) = 3 - 25 = -22\).- The element in the 1st row, 3rd column: \((-1)(-4) + (5)(-5) = 4 - 25 = -21\).- The element in the 1st row, 4th column: \((-1)(-4) + (5)(-1) = 4 - 5 = -1\).- The element in the 2nd row, 1st column: \((6)(5) + (7)(-2) = 30 - 14 = 16\).- The element in the 2nd row, 2nd column: \((6)(-3) + (7)(-5) = -18 - 35 = -53\).- The element in the 2nd row, 3rd column: \((6)(-4) + (7)(-5) = -24 - 35 = -59\).- The element in the 2nd row, 4th column: \((6)(-4) + (7)(-1) = -24 - 7 = -31\).The product matrix \( AB \) is: \[ \begin{bmatrix} -15 & -22 & -21 & -1 \ 16 & -53 & -59 & -31 \end{bmatrix} \]

Conclusion

To summarize: The dimensions of matrix \( A \) are \( 2 \times 2 \), and matrix \( B \) are \( 2 \times 4 \). The product \( AB \) is possible and has dimensions \( 2 \times 4 \). However, \( BA \) is not possible due to mismatched dimensions.

Key concepts

Matrix Dimensions

In matrix mathematics, understanding dimensions is crucial. The dimensions refer to the number of rows and columns in a matrix. This information is important because it determines whether matrices can be multiplied and what the resultant size will be.
When we say a matrix is "m x n", "m" represents the number of rows, and "n" represents the number of columns. For example, Matrix \( A \), defined as \( \begin{bmatrix} -1 & 5 \ 6 & 7 \end{bmatrix} \), has 2 rows and 2 columns, meaning it has dimensions of \( 2 \times 2 \). Similarly, Matrix \( B \), given as \( \begin{bmatrix} 5 & -3 & -4 & -4 \ -2 & -5 & -5 & -1 \end{bmatrix} \), has 2 rows and 4 columns, resulting in dimensions of \( 2 \times 4 \).
Matrix dimensions are more than just a count of rows and columns; they dictate the feasibility of matrix operations such as multiplication. When you're checking whether two matrices can be multiplied, you need to compare these dimensions directly.

Matrix Algebra

Matrix algebra involves performing operations on matrices, such as addition, subtraction, and multiplication. The focus here is on multiplication, which has specific rules regarding dimensions.
Two matrices, say \( A \) and \( B \), can be multiplied if the number of columns in \( A \) matches the number of rows in \( B \). This rule ensures that the elements align properly for multiplication and summation.

• In the given exercise, matrix \( A \) is \( 2 \times 2 \) and matrix \( B \) is \( 2 \times 4 \). For \( AB \), the columns of \( A \) (2) match the rows of \( B \) (2), making \( AB \) possible.
• Conversely, for \( BA \), the columns of \( B \) (4) do not match the rows of \( A \) (2), so \( BA \) is not computable.

Matrix algebra principles allow us to not only calculate products when feasible but also understand why some operations cannot be performed.

Matrix Products

Matrix product computation requires careful multiplication and addition. For each position in the resulting matrix, you take a row from the first matrix and a column from the second matrix to produce a specific result. This involves:

• Multiplying corresponding elements from the row of the first matrix and the column of the second matrix.
• Summing these products to get the final value for that position in the resulting matrix.

For instance, with matrices \( A \) and \( B \) given, the product \( AB \) is calculated by:- Constructing each element. For example, the first element (1st row, 1st column) is: \((-1)(5) + (5)(-2) = -15\).- Repeating this process for each element in the resulting matrix of dimensions \( 2 \times 4 \).The computed \( AB \) matrix is:\[ \begin{bmatrix} -15 & -22 & -21 & -1 \ 16 & -53 & -59 & -31 \end{bmatrix} \]Remember, the resulting matrix from multiplying \( A \) (\( 2 \times 2 \)) by \( B \) (\( 2 \times 4 \)) naturally becomes a \( 2 \times 4 \) matrix itself. These rules reinforce the structured nature of matrix products and emphasize why understanding dimension matching is so important.