Problem 38 - College Algebra

Question

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} 4 & 0 \\ -3 & 5 \\ 0 & 1 \end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\ -2 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right] $$ $$ 5 C-2 B $$

Step-by-Step Solution

Answer

The resulting matrix of the operation \(5C - 2B\) is:\[\left[\begin{array}{rr} -5 & -7 \1 & 9 \end{array}\right]\]

Step 1: Perform Scalar Multiplication

Multiply matrix C by the scalar 5 and matrix B by the scalar 2. This can be done by multiplying each element of the matrices by the corresponding scalar. The resulting matrices are known as the scalar multiples of the original matrices.\nScalable Multiplication yields: \[5C = \left[\begin{array}{rr} 5(1) & 5(-1) \ 5(-1) & 5(1) \end{array}\right] = \left[\begin{array}{rr} 5 & -5 \-5 & 5 \end{array}\right]\]and \[2B = \left[\begin{array}{rr} 2(5) & 2(1) \ 2(-2) & 2(-2) \end{array}\right] = \left[\begin{array}{rr} 10 & 2 \-4 & -4 \end{array}\right]\]

Step 2: Perform the Matrix Subtraction

Subtract the scalar multiple of matrix B from the scalar multiple of matrix C. This is done by subtracting the corresponding elements in each matrix.\[5C - 2B = \left[\begin{array}{rr} 5 & -5 \-5 & 5 \end{array}\right] - \left[\begin{array}{rr} 10 & 2 \-4 & -4 \end{array}\right] = \left[\begin{array}{rr} 5-10 & -5-2 \-5-(-4) & 5-(-4)\end{array}\right]\]

Key Concepts

Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a single number, known as the scalar. For example, let's consider matrix \( C \) which is \[ C = \begin{bmatrix} 1 & -1 \ -1 & 1 \end{bmatrix} \] and the scalar 5. To perform scalar multiplication, you multiply each element of \( C \) by 5. Here’s how it works:

• The first element, 1, becomes \( 5 \times 1 = 5 \).
• The second element, -1, becomes \( 5 \times -1 = -5 \).
• Continue this for all elements in the matrix.

Thus, multiplying the whole matrix \( C \) by 5 results in \[ 5C = \begin{bmatrix} 5 & -5 \ -5 & 5 \end{bmatrix} \].
This operation is crucial because it allows us to scale matrices, either increasing or decreasing their magnitude by a uniform factor, which is the scalar.

Matrix Subtraction

Matrix subtraction involves subtracting corresponding elements of two matrices of the same dimensions. For this operation to be possible, both matrices must have identical dimensions. Consider two matrices \( A \) and \( B \), with the same number of rows and columns. Here’s how matrix subtraction works:

• Subtract the first element of matrix \( B \) from the first element of matrix \( A \).
• Continue this process for each corresponding element within the matrices.

Using our exercise as an example, we subtract \( 2B \) from \( 5C \):\[5C - 2B = \begin{bmatrix} 5 & -5 \ -5 & 5 \end{bmatrix} - \begin{bmatrix} 10 & 2 \ -4 & -4 \end{bmatrix} = \begin{bmatrix} 5 - 10 & -5 - 2 \-5 - (-4) & 5 - (-4) \end{bmatrix} \]
In subtraction, it's vital to align elements correctly and ensure both matrices have matching sizes.

Matrix Addition

Matrix addition is a simple yet powerful operation in matrix algebra. Just like matrix subtraction, addition requires the matrices involved to have the same dimensions. This ensures that each element from the first matrix has a corresponding pair in the second matrix. When adding matrices:

• Add each corresponding pair of elements together.
• The result is a new matrix of the same dimensions.

For instance, if two matrices \( X \) and \( Y \) are both 2x2:\[X = \begin{bmatrix} a & b \c & d \end{bmatrix}, Y = \begin{bmatrix} e & f \g & h \end{bmatrix} \]Matrix addition would look like:\[X + Y = \begin{bmatrix} a+e & b+f \c+g & d+h \end{bmatrix}\]
This process is quite straightforward. Ensure the matrices are the same size before attempting addition.

Matrix Dimensions

Understanding matrix dimensions is fundamental to performing matrix operations like addition, subtraction, and multiplication. A matrix's dimensions are defined by the number of rows (horizontal lines of elements) and columns (vertical lines of elements) it contains.
For example, a matrix with 3 rows and 2 columns is described as a 3x2 matrix, denoted as:\[\begin{bmatrix} a_{11} & a_{12} \a_{21} & a_{22} \a_{31} & a_{32} \end{bmatrix} \]
Knowing the dimensions is crucial because it determines which operations can be performed.

• For scalar multiplication, any matrix dimensions are permissible.
• For addition and subtraction, matrices must have the same dimensions.
• For matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix.

Grasping these basic principles of matrix dimensions is key to mastering more complex matrix operations.