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. $$ \begin{array}{l} A=\left[\begin{array}{ccc} -4 & -1 & 3 \\ 2 & -3 & 5 \\ 1 & 5 & 3 \end{array}\right] \\ B=\left[\begin{array}{ccc} -2 & 4 & 3 \\ -1 & 1 & -1 \\ 4 & 0 & 2 \end{array}\right] \end{array} $$

Short answer

AB and BA are both defined and their products are 3x3 matrices.

Step-by-step solution

Determine Dimensions of Matrices

Matrix \(A\) is a 3x3 matrix because it has 3 rows and 3 columns. Similarly, matrix \(B\) is a 3x3 matrix, also with 3 rows and 3 columns. Both matrices are square and of the same size.

Compatibility Check for Multiplication

Since both matrices \(A\) and \(B\) are 3x3, they can be multiplied in both orders: \(AB\) and \(BA\). The resulting dimensions for both products will be 3x3, as the number of columns in the first matrix matches the number of rows in the second matrix for both products.

Calculate Product AB

To find \(AB\), multiply each element of the rows of matrix \(A\) by the corresponding elements of the columns of matrix \(B\) and sum them up.- First row, first column: \((-4)(-2) + (-1)(-1) + (3)(4) = 8 + 1 + 12 = 21\)- First row, second column: \((-4)(4) + (-1)(1) + (3)(0) = -16 - 1 + 0 = -17\)- First row, third column: \((-4)(3) + (-1)(-1) + (3)(2) = -12 + 1 + 6 = -5\)Continuing in this manner for all elements:\[AB = \begin{bmatrix}21 & -17 & -5 \32 & -1 & 5 \6 & 9 & 8\end{bmatrix}\]

Calculate Product BA

To compute \(BA\), use a similar process:- First row, first column: \((-2)(-4) + (4)(2) + (3)(1) = 8 + 8 + 3 = 19\)- First row, second column: \((-2)(-1) + (4)(-3) + (3)(5) = 2 - 12 + 15 = 5\)- First row, third column: \((-2)(3) + (4)(5) + (3)(3) = -6 + 20 + 9 = 23\)Continuing for each element:\[BA = \begin{bmatrix}19 & 5 & 23 \5 & -11 & 4 \-11 & -23 & 29\end{bmatrix}\]

Verify and Conclude

Both matrix products were calculated by summing the respective element-wise products. If any errors are found, recheck calculations, otherwise conclude with final matrices:\[AB = \begin{bmatrix} 21 & -17 & -5 \ 32 & -1 & 5 \ 6 & 9 & 8 \end{bmatrix}\]\[BA = \begin{bmatrix} 19 & 5 & 23 \ 5 & -11 & 4 \ -11 & -23 & 29 \end{bmatrix}\]

Key concepts

matrix dimensions

Matrix dimensions are an essential aspect of understanding matrices and their properties. The dimension of a matrix is expressed as 'rows' by 'columns'. For example, in the provided exercise, both matrices \( A \) and \( B \) are stated to be \( 3 \times 3 \) matrices. This means each matrix has 3 rows and 3 columns.

The significance of matrix dimensions goes beyond simply counting rows and columns. It is crucial when considering matrix operations like addition, subtraction, and particularly, multiplication. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix. For both matrix \( A \) and matrix \( B \), since they are \( 3 \times 3 \) matrices, they can be multiplied in both orders—\( AB \) and \( BA \). The resultant matrix from both operations will also be \( 3 \times 3 \).

Understanding matrix dimensions helps in visualizing the structure and compatibility of matrices, setting a foundational skill for executing further mathematical operations effectively.

square matrices

Square matrices are matrices with an equal number of rows and columns. In mathematics, square matrices are especially interesting due to their unique properties and their frequent use in more advanced topics like determinants, inverses, and eigenvalues.

In the exercise at hand, both matrices \( A \) and \( B \) are \( 3 \times 3 \) square matrices. These matrices hold particular significance in linear algebra since they can sometimes be inverted if they’re non-singular, meaning their determinant is not zero. This is important when solving systems of equations using matrices.

Square matrices also have special properties when it involves certain operations. For example, when square matrices of the same size are multiplied, they result in another square matrix. This characteristic is evident in both the calculated products \( AB \) and \( BA \), where both operations produced another \( 3 \times 3 \) square matrix.

Recognizing the form of square matrices is pivotal when embarking on various matrix operations, ensuring that computations like those demonstrated in the exercise are performed successfully.

matrix products

Matrix product or matrix multiplication is a fundamental operation in linear algebra. It involves combining two matrices to produce a third matrix. The rule for multiplying these involves taking each row element of the first matrix and multiplying it by the corresponding column elements of the second matrix, followed by summing these products.

For instance, in the problem solution, the first element of the product \( AB \) is obtained by multiplying elements of the first row of matrix \( A \) by elements of the first column of matrix \( B \), which gets summed up to form a single element in the resulting matrix. This is repeated for each combination of rows and columns to fill the entire matrix.

An interesting feature of matrix multiplication is that it is not commutative, meaning \( AB \) does not necessarily equal \( BA \). In the exercise, this is demonstrated by different results for the matrices \( AB \) and \( BA \). Such nuances are key in various applications like transformations and solving matrix equations.

Understanding matrix products extends to many practical applications in fields like computer graphics, physics simulations, and data transformations, making it a vital skill in mathematics and related disciplines.