Problem 23 - Precalculus Mathematics for Calculus

Question

The matrices \(A, B, C, D, E, F, G\) and \(H\) are defined as follows. $$\begin{aligned} &A=\left[\begin{array}{rr} 2 & -5 \\ 0 & 7 \end{array}\right] \quad B=\left[\begin{array}{rrr} 3 & \frac{1}{2} & 5 \\ 1 & -1 & 3 \end{array}\right] \quad C=\left[\begin{array}{rrr} 2 & -\frac{5}{2} & 0 \\ 0 & 2 & -3 \end{array}\right]\\\ &D=\left[\begin{array}{llll} 7 & 3 \end{array}\right] \quad E=\left[\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right] \quad F=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\\\ &G=\left[\begin{array}{rrr} 5 & -3 & 10 \\ 6 & 1 & 0 \\ -5 & 2 & 2 \end{array}\right] \quad H=\left[\begin{array}{rr} 3 & 1 \\ 2 & -1 \end{array}\right] \end{aligned}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. (a) \(5 A\) (b) \(C-5 A\)

Step-by-Step Solution

Answer

(a) \(5A = \begin{bmatrix} 10 & -25 \\ 0 & 35 \end{bmatrix}\); (b) Cannot be performed.

Step 1: Multiply Matrix A by Scalar 5

To find \(5A\), we multiply every element in matrix \(A\) by the scalar 5. Matrix \(A\) is \(\begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\). Calculate each element:- The element in the first row, first column: \(5 \times 2 = 10\).- The element in the first row, second column: \(5 \times (-5) = -25\).- The element in the second row, first column: \(5 \times 0 = 0\).- The element in the second row, second column: \(5 \times 7 = 35\).Thus, \(5A = \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix}\).

Step 2: Explain Subtraction of Matrices C and A

Matrix \(C\) is \(\begin{bmatrix} 2 & -\frac{5}{2} & 0 \ 0 & 2 & -3 \end{bmatrix}\) which has a size of 2x3, and matrix \(5A = \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix}\) which has a size of 2x2. To subtract two matrices, they must be of the same dimension.Since the dimensions of matrices \(C\) (2x3) and \(5A\) (2x2) aren't the same, the subtraction \(C - 5A\) cannot be performed.

Key Concepts

Matrix Scalar Multiplication

Matrix scalar multiplication is a straightforward operation when dealing with matrices. This operation involves taking a scalar, which is essentially a single number, and multiplying it by every element in the matrix.

• Process: To perform scalar multiplication, apply the scalar to each element of the matrix. For example, in the given problem, multiplying matrix \(A\) by the scalar 5 means every element of \(A\) is multiplied by 5.
• Example: Consider matrix \(A\):\[A = \begin{bmatrix} 2 & -5 \ 0 & 7 \end{bmatrix}\]Multiplying by scalar 5, each element becomes:\[5A = \begin{bmatrix} 5 \times 2 & 5 \times -5 \ 5 \times 0 & 5 \times 7 \end{bmatrix} = \begin{bmatrix} 10 & -25 \ 0 & 35 \end{bmatrix}\]

Scalar multiplication does not require the matrix to have certain dimensions, as it applies uniformly across the entire matrix. It simplifies many calculations in linear algebra and is commonly used in operations involving matrices like scaling and transformations.
Understanding scalar multiplication is foundational for more complex operations involving matrices.

Matrix Subtraction

Matrix subtraction involves taking two matrices and subtracting the elements of one matrix from the corresponding elements of another. However, this operation requires that both matrices have the exact same dimensions. If the matrices do not match in size, subtraction cannot proceed.

• Requirements: The matrices must have the same number of rows and columns. This is because you are subtracting corresponding elements.
• Illustration: Suppose \[X = \begin{bmatrix} x_{11} & x_{12} \ x_{21} & x_{22} \end{bmatrix}\]and \[Y = \begin{bmatrix} y_{11} & y_{12} \ y_{21} & y_{22} \end{bmatrix}\]Then, \[X - Y = \begin{bmatrix} x_{11} - y_{11} & x_{12} - y_{12} \ x_{21} - y_{21} & x_{22} - y_{22} \end{bmatrix}\]

This principle is why in the original exercise, subtraction between matrices \(C\) and \(5A\) is not possible, as \(C\) is a 2x3 matrix and \(5A\) is a 2x2 matrix. They simply don't have compatible dimensions for subtraction.
To perform matrix subtraction, always ensure that both matrices are of the same size to allow a one-to-one element correspondence.

Matrix Dimensions

Matrix dimensions, or sizes, are crucial when performing operations such as addition, subtraction, or even some forms of multiplication. Dimensions are indicated as rows by columns (e.g., 2x3, which means 2 rows and 3 columns). Understanding matrix dimensions is essential for ensuring that operations are valid and can be executed.

• Notation: The order of dimensions is expressed as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.
• Importance: Correct dimensions are necessary for operations like addition and subtraction, where you need the same dimensions; and for specific multiplication methods, such as matrix-matrix multiplication, where the number of columns in the first matrix must equal the number of rows in the second matrix.

For example, matrix \(F\) in the exercise is a classic 3x3 identity matrix. Knowing its dimensions instantly informs us about potential compatible operations or transformations.
In the original problem, the inability to subtract \(C\) and \(5A\) is due to their differing dimensions. Recognizing the dimensions and how they affect operations is a foundational concept in mastering matrix algebra.