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. $$ \begin{array}{l} A=\left[\begin{array}{cc} 2 & 6 \\ 6 & 2 \\ 5 & -1 \end{array}\right] \\ B=\left[\begin{array}{ccc} -4 & 5 & 0 \\ -4 & 4 & -4 \end{array}\right] \end{array} $$

Short answer

The product \( AB \) is possible with dimensions \( 3 \times 3 \), and \( BA \) does not exist.

Step-by-step solution

Determine the Dimensions of Matrix A

Matrix \( A \) is given as \( \begin{bmatrix} 2 & 6 \ 6 & 2 \ 5 & -1 \end{bmatrix} \). It has 3 rows and 2 columns. Therefore, the dimensions of matrix \( A \) are \( 3 \times 2 \).

Determine the Dimensions of Matrix B

Matrix \( B \) is given as \( \begin{bmatrix} -4 & 5 & 0 \ -4 & 4 & -4 \end{bmatrix} \). It has 2 rows and 3 columns. Therefore, the dimensions of matrix \( B \) are \( 2 \times 3 \).

Check Dimensions for AB and BA

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. For \( AB \), matrix \( A \) is \( 3 \times 2 \) and matrix \( B \) is \( 2 \times 3 \), so the multiplication is possible and the resulting matrix \( AB \) will have dimensions \( 3 \times 3 \).\For \( BA \), matrix \( B \) is \( 2 \times 3 \) and matrix \( A \) is \( 3 \times 2 \), so multiplication is not possible.

Calculate the Product AB

To find \( AB \), multiply each element of the rows of \( A \) with corresponding elements of the columns of \( B \) and sum them up.\\((AB)_{11} = 2(-4) + 6(-4) = -8 - 24 = -32\)\\((AB)_{12} = 2(5) + 6(4) = 10 + 24 = 34\)\\((AB)_{13} = 2(0) + 6(-4) = 0 - 24 = -24\)\\((AB)_{21} = 6(-4) + 2(-4) = -24 - 8 = -32\)\\((AB)_{22} = 6(5) + 2(4) = 30 + 8 = 38\)\\((AB)_{23} = 6(0) + 2(-4) = 0 - 8 = -8\)\\((AB)_{31} = 5(-4) + (-1)(-4) = -20 + 4 = -16\)\\((AB)_{32} = 5(5) + (-1)(4) = 25 - 4 = 21\)\\((AB)_{33} = 5(0) + (-1)(-4) = 0 + 4 = 4\)\So, \( AB = \begin{bmatrix} -32 & 34 & -24 \ -32 & 38 & -8 \ -16 & 21 & 4 \end{bmatrix} \).

State the Non-existence of BA

Since the number of columns in \( B \) (3) is not equal to the number of rows in \( A \) (3), the product \( BA \) does not exist.

Key concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when working with matrices, especially for matrix multiplication. Each matrix is defined by its dimensions, often written as "rows x columns." This tells us the number of rows and columns a matrix contains. For example, if a matrix has three rows and two columns, its dimensions are expressed as \(3 \times 2\).

To determine if two matrices can be multiplied, we need to check their dimensions. Specifically, the number of columns in the first matrix must match the number of rows in the second matrix. This alignment allows for the multiplication process, where each element from a row in the first matrix interacts with a corresponding element in a column in the second matrix.

In the given example, Matrix \(A\) is \(3 \times 2\) and Matrix \(B\) is \(2 \times 3\). This means that the inner numbers (2 and 2) align, allowing for the possible multiplication of \(A\) and \(B\). The resulting product \(AB\) will have dimensions determined by the outer numbers, resulting in a \(3 \times 3\) matrix.

Matrix Product

The matrix product, or the result of multiplying two matrices, is achieved by performing a series of dot products between rows of the first matrix and columns of the second matrix. This process involves multiplying corresponding elements and summing them up to get each element of the resulting matrix.

For our specific matrices \(A\) and \(B\), the product \(AB\) was calculated by taking each row of matrix \(A\) and pairing it with each column of matrix \(B\). For instance, to find \((AB)_{11}\), or the element in the first row and first column of the product matrix, each element of the first row of \(A\) is multiplied by the corresponding element of the first column of \(B\) and summed:

• \(2(-4) + 6(-4) = -32\)

This process is repeated for each element of the product matrix \(AB\), resulting in a new set of values after applying this multiplication and addition process across all row-column combinations. This detailed sequence ensures that each element in the product matrix accurately represents the cumulative interactions between the respective rows and columns of the original matrices.

Non-Commutative Multiplication

Matrix multiplication is non-commutative, meaning that the order of multiplication affects the result. In simpler terms, \(AB\) might not equal \(BA\).

As illustrated in the exercise, multiplying matrix \(A\) by \(B\) (resulting in \(AB\)) is possible due to the compatible dimensions (where the number of columns in \(A\) matches the number of rows in \(B\)). However, trying to reverse the order by computing \(BA\) is not possible here. This is because matrix \(B\) has more columns (3) than there are rows in \(A\) (also 3), which breaks the requirement of matching dimensions for multiplication.

This non-commutative property is a fundamental aspect of matrix operations that distinguishes them from scalar arithmetic (like regular numbers), making it important to always verify dimensions before attempting to multiply matrices. Understanding this concept helps in avoiding potential errors and correctly performing matrix multiplications in various applications like computer graphics, physics simulations, and system equations.