Problem 37 - Precalculus: Functions and Graphs

Question

Let $$A=\left[\begin{array}{rrr}3 & -3 & 7 \\\2 & 6 & -2 \\\4 & 2 & 5\end{array}\right] \text { and }B=\left[\begin{array}{rrr}-9 & 5 & -8 \\\3 & -7 & 1 \\\\-1 & 2 & 6\end{array}\right]$$ Evaluate the matrix expression. $$A^{2}-5 B$$

Step-by-Step Solution

Answer

The result of \(A^2 - 5B\) is \[ \begin{bmatrix} 92 & -52 & 70 \\ 11 & 69 & -15 \\ 43 & -12 & 12 \end{bmatrix} \]

Step 1: Calculate the square of matrix A

To find \(A^2\), we need to multiply matrix A by itself. Let's set up the multiplication:\[A imes A = \begin{bmatrix}3 & -3 & 7 \2 & 6 & -2 \4 & 2 & 5\end{bmatrix}\times\begin{bmatrix}3 & -3 & 7 \2 & 6 & -2 \4 & 2 & 5\end{bmatrix}\]1. Compute each element of the resulting matrix by taking dot products of the appropriate row of the first matrix with the appropriate column of the second matrix. After completing calculations, you obtain:\[A^2 = \begin{bmatrix} 47 & -27 & 30 \ 26 & 34 & -10 \ 38 & -2 & 42 \end{bmatrix}\]

Step 2: Multiply Matrix B by 5

To calculate \(5B\), multiply each element of matrix B by 5:\[ 5B = 5 \times \begin{bmatrix} -9 & 5 & -8 \ 3 & -7 & 1 \ -1 & 2 & 6 \end{bmatrix} = \begin{bmatrix} -45 & 25 & -40 \ 15 & -35 & 5 \ -5 & 10 & 30 \end{bmatrix} \]

Step 3: Subtract 5B from A^2

Finally, to evaluate the expression \(A^2 - 5B\), subtract the corresponding elements of matrix \(5B\) from matrix \(A^2\):\[A^2 - 5B = \begin{bmatrix} 47 & -27 & 30 \ 26 & 34 & -10 \ 38 & -2 & 42 \end{bmatrix} -\begin{bmatrix} -45 & 25 & -40 \ 15 & -35 & 5 \ -5 & 10 & 30 \end{bmatrix}= \begin{bmatrix} 92 & -52 & 70 \ 11 & 69 & -15 \ 43 & -12 & 12 \end{bmatrix}\]

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra, crucial for various applications in mathematics and related fields. Unlike regular multiplication with numbers, matrix multiplication involves a specific process:

• Two matrices can be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix.
• The element in the resulting matrix is found by taking the dot product of rows from the first matrix and columns from the second matrix.

For instance, given a matrix \( A \) of size \( m \times n \) and matrix \( B \) of size \( n \times p \), the resulting matrix \( C = A \times B \) will be of size \( m \times p \). Each element \( c_{ij} \) is calculated as:\[c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \, \ldots \, + a_{in}b_{nj}\]This operation is not commutative, meaning \( A \times B eq B \times A \). Thus, the order of multiplication matters significantly.
When squaring a matrix like \( A^2 \), it means \( A \times A \), ensuring to follow the same rules.

Matrix Subtraction

Matrix subtraction, like addition, requires both matrices to have the same dimensions. This process involves subtracting corresponding elements of each matrix to produce a new matrix of the same size.

• The matrices involved must have identical structures, that is, the same number of rows and columns.
• Subtraction is performed element by element.

For example, given two matrices \( C \) and \( D \) of the same size, the subtraction \( C - D \) results in a new matrix \( E \) where each element \( e_{ij} = c_{ij} - d_{ij} \).
Matrix subtraction is straightforward and follows simple arithmetic rules applied consistently across all corresponding elements.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a scalar, which is a single number, resulting in a new matrix where each element is scaled by the scalar value.

• Scalar multiplication affects each entry of the matrix independently.
• The resulting matrix maintains the same dimensions as the original matrix.

If matrix \( F \) has elements \( f_{ij} \), and it is multiplied by a scalar \( k \), the resulting matrix \( kF \) has elements \( kf_{ij} \).
This type of multiplication is useful for adjusting the magnitude of matrices, such as in operations like computing \( 5B \), where each element of \( B \) is multiplied by 5.

Matrices

Matrices are a key component in many areas of mathematics and science, serving as a way to organize and manage data or perform computations. A matrix is a rectangular array of numbers arranged in rows and columns.

• Matrices can vary in size, such as 2x2, 3x3, or even larger, with each dimension indicating the number of rows and columns.
• Matrices are used in solving systems of equations, transformations in geometry, and many other practical applications.

Each component of a matrix is called an element, typically denoted as \( a_{ij} \), where \( i \) represents the row number, and \( j \) represents the column number.
Understanding matrices is essential as they form the basis for matrix operations including addition, subtraction, multiplication, and more complex transformations.