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. $$ A=\left[\begin{array}{ll} 3 & 7 \\ 2 & 5 \end{array}\right] B=\left[\begin{array}{ll} 1 & -1 \\ 3 & -3 \end{array}\right] $$

Short answer

(a) AB and BA both have dimensions 2x2. (b) AB = [[24, -24], [17, -17]], BA = [[1, 2], [3, 6]].

Step-by-step solution

Identify Dimensions

Matrix \(A\) has 2 rows and 2 columns, so its dimensions are \(2 \times 2\). Similarly, matrix \(B\) also has 2 rows and 2 columns, so its dimensions are \(2 \times 2\). Since both matrices are \(2 \times 2\), they can be multiplied in either order.

Determine Resultant Dimensions

Since both matrices \(A\) and \(B\) are \(2 \times 2\), the resultant matrices, when multiplied in either order (\(AB\) and \(BA\)), will also be \(2 \times 2\).

Calculate AB

To find \(AB\), we multiply each row of \(A\) by each column of \(B\):\[AB = \begin{bmatrix} 3 & 7 \ 2 & 5 \end{bmatrix} \begin{bmatrix} 1 & -1 \ 3 & -3 \end{bmatrix}\]Compute each element: - First row, first column: \(3\times1 + 7\times3 = 3 + 21 = 24\) - First row, second column: \(3\times(-1) + 7\times(-3) = -3 - 21 = -24\) - Second row, first column: \(2\times1 + 5\times3 = 2 + 15 = 17\) - Second row, second column: \(2\times(-1) + 5\times(-3) = -2 - 15 = -17\)The product \(AB\) is \[\begin{bmatrix} 24 & -24 \ 17 & -17 \end{bmatrix}\].

Calculate BA

To find \(BA\), we multiply each row of \(B\) by each column of \(A\):\[BA = \begin{bmatrix} 1 & -1 \ 3 & -3 \end{bmatrix} \begin{bmatrix} 3 & 7 \ 2 & 5 \end{bmatrix}\]Compute each element: - First row, first column: \(1\times3 + (-1)\times2 = 3 - 2 = 1\) - First row, second column: \(1\times7 + (-1)\times5 = 7 - 5 = 2\) - Second row, first column: \(3\times3 + (-3)\times2 = 9 - 6 = 3\) - Second row, second column: \(3\times7 + (-3)\times5 = 21 - 15 = 6\)The product \(BA\) is \[\begin{bmatrix} 1 & 2 \ 3 & 6 \end{bmatrix}\].

Key concepts

Matrix Dimensions

In the world of matrices, understanding the concept of matrix dimensions is crucial. The dimensions of a matrix tell us the number of rows and columns it contains. For instance, if a matrix has 3 rows and 4 columns, its dimensions are expressed as "3 x 4". In our exercise, both the given matrices, \( A \) and \( B \), have dimensions \( 2 \times 2 \). This means each matrix has 2 rows and 2 columns. Knowing the dimensions helps us determine whether matrix multiplication is possible. For multiplication to occur, the number of columns in the first matrix must equal the number of rows in the second matrix. Fortunately, with \( 2 \times 2 \) matrices like \( A \) and \( B \), multiplication in either order, \( AB \) or \( BA \), is possible. This is an important step in preparing for matrix multiplication.

Resultant Matrix

Once we know that two matrices can be multiplied, the next step is to determine the dimensions of the resultant matrix. This matrix is the product of the two multiplied matrices. For multiplication of an \( m \times n \) matrix with an \( n \times p \) matrix, the resultant matrix will have the dimensions \( m \times p \). In our case, both matrices \( A \) and \( B \) are \( 2 \times 2 \), leading to a resultant matrix \( AB \) or \( BA \) that is also \( 2 \times 2 \). Therefore, the product of two square matrices of the same dimension is always another square matrix of that dimension. This unchanging size is a characteristic of square matrices when they are multiplied together.

Matrix Products

Matrix multiplication involves a specific method of calculating the product matrix. It's not merely multiplying corresponding elements, but rather:

• Each element of the resulting matrix is a sum of products.
• The elements in a row of the first matrix are multiplied by the corresponding elements in a column of the second matrix, and those products are summed.

For example, to compute the product \( AB \), take each row of \( A \) and multiply it with each column of \( B \). In our exercise, the element in the first row, first column of \( AB \) is calculated as: \( 3\times1 + 7\times3 = 24 \). This process is repeated for each element in the resultant matrix, making matrix multiplication a meticulous but straightforward process. Remember, the order of multiplication impacts the result, meaning \( AB \) doesn't always equal \( BA \). This is a unique property of matrix multiplication compared to numerical multiplication.