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. $$ \begin{array}{l} A=\left[\begin{array}{cc} 0 & 1 \\ 1 & -1 \\ -2 & -4 \end{array}\right] \\ B=\left[\begin{array}{cc} -2 & 0 \\ 3 & 8 \end{array}\right] \end{array} $$

Short answer

\(AB = \begin{bmatrix} 3 & 8 \\ -5 & -8 \\ -8 & -32 \end{bmatrix}\); \(BA\) is not possible.

Step-by-step solution

Determine the Dimensions of Matrices

Matrix \( A \) is defined as \( 3 \times 2 \), because it has 3 rows and 2 columns. Matrix \( B \) is \( 2 \times 2 \), since it has 2 rows and 2 columns. To find the product \( AB \), the number of columns in \( A \) must match the number of rows in \( B \). Thus, \( AB \) is possible, and its dimension will be \( 3 \times 2 \). To find the product \( BA \), the number of columns in \( B \) must match the number of rows in \( A \), which does not hold. So, \( BA \) is not possible.

Compute the Product AB

The product \( AB \) is calculated as follows: 1. Multiply each element of the rows of \( A \) with the corresponding element of the columns of \( B \) and sum them up.\[AB = \begin{bmatrix}0 \cdot (-2) + 1 \cdot 3 & 0 \cdot 0 + 1 \cdot 8\1 \cdot (-2) + (-1) \cdot 3 & 1 \cdot 0 + (-1) \cdot 8\-2 \cdot (-2) + (-4) \cdot 3 & -2 \cdot 0 + (-4) \cdot 8\end{bmatrix}\]\[= \begin{bmatrix}3 & 8\-2 - 3 & 0 - 8\4 - 12 & 0 - 32\end{bmatrix}\]\[= \begin{bmatrix}3 & 8\-5 & -8\-8 & -32\end{bmatrix}\]

Conclusion of Computation

Matrix \( AB \) has been computed with dimensions \( 3 \times 2 \) as\[\begin{bmatrix}3 & 8\-5 & -8\-8 & -32\end{bmatrix}\]. The product \( BA \) is not possible due to mismatched dimensions.

Key concepts

Matrix Dimensions

Understanding the dimensions of a matrix is crucial in matrix manipulation, as it determines what operations are possible, particularly matrix multiplication. A matrix's dimensions are given by the number of rows and columns it contains, stated as \( m \times n \), where \( m \) is the number of rows and \( n \) is the number of columns.
For example, if matrix \( A \) has 3 rows and 2 columns, we denote it as a \( 3 \times 2 \) matrix. Similarly, a \( 2 \times 2 \) matrix \( B \) consists of 2 rows and 2 columns.

This matrix dimension rules out some operations depending on whether these dimensions align. When multiplying two matrices together, such as matrices \( A \) and \( B \), the number of columns in the first matrix must equal the number of rows in the second matrix.
If the dimensions of matrix \( A \) are \( 3 \times 2 \) and \( B \) are \( 2 \times 2 \), then the matrix product \( AB \) can be computed, resulting in an output dimension of \( 3 \times 2 \) because the number of rows from \( A \) and columns from \( B \) dictate this.

On the other hand, \( BA \) cannot be formed because the columns of \( B \) (2) do not match the rows of \( A \) (3). This illustrates how matrix dimensions directly impact the feasibility of certain calculations.

Matrix Product

Matrix multiplication, often known as the matrix product, is a fundamental operation in linear algebra. It involves the combination of two matrices to produce a new matrix, following some specific algebraic rules.
To determine the product of matrices \( A \) and \( B \), denoted as \( AB \), each element in the resulting matrix is calculated through the dot product of a row from the first matrix and a column from the second matrix.

Here’s how you do it:

• Take a row from the first matrix \( A \) and a column from the second matrix \( B \).
• Multiply the respective elements pairwise and then sum these products.
• The sum becomes an entry in the resulting matrix at the row and column position being considered.

For example, if element \((1,1)\) in matrix \( AB \) is calculated by the row 1 of \( A \) and column 1 of \( B \).
Applying this to matrices \[ A = \begin{bmatrix} 0 & 1 \ 1 & -1 \ -2 & -4 \end{bmatrix} \] and \[ B = \begin{bmatrix} -2 & 0 \ 3 & 8 \end{bmatrix} \], we follow the steps to find \( AB \):

\( AB = \begin{bmatrix} (0 \cdot -2 + 1 \cdot 3) & (0 \cdot 0 + 1 \cdot 8) \ (1 \cdot -2 + -1 \cdot 3) & (1 \cdot 0 + -1 \cdot 8) \ (-2 \cdot -2 + -4 \cdot 3) & (-2 \cdot 0 + -4 \cdot 8) \end{bmatrix} \)
This computation results in:\[ \begin{bmatrix} 3 & 8 \ -5 & -8 \ 4 - 32 \end{bmatrix} \] which simplifies to \[ \begin{bmatrix} 3 & 8 \ -5 & -8 \ -8 & -32 \end{bmatrix} \].

Matrix Algebra

Matrix algebra is a broad area dealing with the study of matrices and the many operations that can be performed on them, such as addition, subtraction, multiplication, and more. It's essential for understanding complex mathematical concepts, especially those involving systems of equations and transformations.
Matrices have their own set of rules and properties that parallel those we know from regular number algebra, yet they require specific conditions.

Here are some key concepts:

• Addition and Subtraction: Two matrices can only be added or subtracted if they have the same dimensions.
• Multiplication: As seen in our example, matrices can be multiplied only when the number of columns in the first is equal to the number of rows in the second. The resulting matrix adopts the rows of the first and columns of the second.
• Identity Matrices: These are square matrices (same number of rows and columns) with 1s on the diagonal from top left to bottom right and 0s elsewhere. Multiplying any matrix with an identity matrix leaves the original matrix unchanged.

Matrix algebra also involves more advanced topics such as determinants, inverses, and eigenvalues, each helping solve different mathematical problems. Understanding these principles lays the groundwork for more advanced computational and applied work in areas like physics, computer science, and engineering.