Problem 1 - Precalculus: Functions and Graphs

Question

Find, if possible, \(A+B, A-B, 2 A,\) and \(-3 B\). $$A=\left[\begin{array}{rr} 5 & -2 \\ 1 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 4 & 1 \\ -3 & 2 \end{array}\right]$$

Step-by-Step Solution

Answer

\(A+B = \begin{bmatrix} 9 & -1 \\ -2 & 5 \end{bmatrix}\), \(A-B = \begin{bmatrix} 1 & -3 \\ 4 & 1 \end{bmatrix}\), \(2A = \begin{bmatrix} 10 & -4 \\ 2 & 6 \end{bmatrix}\), \(-3B = \begin{bmatrix} -12 & -3 \\ 9 & -6 \end{bmatrix}\).

Step 1: Check Matrix Dimensions

Before performing any operations, ensure that the matrices \(A\) and \(B\) are of the same dimensions. Matrix \(A\) is a \(2 \times 2\) matrix, and matrix \(B\) is also a \(2 \times 2\) matrix. Thus, they can be added or subtracted.

Step 2: Calculate \(A+B\)

Add the corresponding elements of matrices \(A\) and \(B\).\[A + B = \begin{bmatrix} 5 & -2 \ 1 & 3 \end{bmatrix} + \begin{bmatrix} 4 & 1 \ -3 & 2 \end{bmatrix} = \begin{bmatrix} 5+4 & -2+1 \ 1+(-3) & 3+2 \end{bmatrix} = \begin{bmatrix} 9 & -1 \ -2 & 5 \end{bmatrix}\]

Step 3: Calculate \(A-B\)

Subtract the corresponding elements of matrix \(B\) from matrix \(A\).\[A - B = \begin{bmatrix} 5 & -2 \ 1 & 3 \end{bmatrix} - \begin{bmatrix} 4 & 1 \ -3 & 2 \end{bmatrix} = \begin{bmatrix} 5-4 & -2-1 \ 1-(-3) & 3-2 \end{bmatrix} = \begin{bmatrix} 1 & -3 \ 4 & 1 \end{bmatrix}\]

Step 4: Calculate \(2A\)

Multiply each element of matrix \(A\) by 2.\[2A = 2 \times \begin{bmatrix} 5 & -2 \ 1 & 3 \end{bmatrix} = \begin{bmatrix} 2\times5 & 2\times(-2) \ 2\times1 & 2\times3 \end{bmatrix} = \begin{bmatrix} 10 & -4 \ 2 & 6 \end{bmatrix}\]

Step 5: Calculate \(-3B\)

Multiply each element of matrix \(B\) by -3.\[-3B = -3 \times \begin{bmatrix} 4 & 1 \ -3 & 2 \end{bmatrix} = \begin{bmatrix} -3\times4 & -3\times1 \ -3\times(-3) & -3\times2 \end{bmatrix} = \begin{bmatrix} -12 & -3 \ 9 & -6 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is one of the fundamental operations you can perform on matrices. To add two matrices, they must have the same dimensions, meaning the same number of rows and columns. This ensures each element in one matrix has a corresponding element in the other.

Matrix addition involves adding corresponding elements from each matrix. For example, consider matrices \( A \) and \( B \):

• \( A = \begin{bmatrix} 5 & -2 \ 1 & 3 \end{bmatrix} \)
• \( B = \begin{bmatrix} 4 & 1 \ -3 & 2 \end{bmatrix} \)

To find \( A + B \), you simply add each pair of corresponding elements:

• Top left element from \( A \) and \( B \): \( 5 + 4 = 9 \)
• Top right elements: \( -2 + 1 = -1 \)
• Bottom left elements: \( 1 + (-3) = -2 \)
• Bottom right elements: \( 3 + 2 = 5 \)

The result of \( A + B \) is \( \begin{bmatrix} 9 & -1 \ -2 & 5 \end{bmatrix} \). Simply put, you merge the matrices by adding corresponding values together.

Matrix Subtraction

Matrix subtraction is quite similar to matrix addition, but instead, you subtract corresponding elements. Again, it's necessary that both matrices share the same dimensions to perform this operation.

Let's subtract matrix \( B \) from matrix \( A \):

• \( A = \begin{bmatrix} 5 & -2 \ 1 & 3 \end{bmatrix} \)
• \( B = \begin{bmatrix} 4 & 1 \ -3 & 2 \end{bmatrix} \)

When you perform \( A - B \), subtract each element in \( B \) from the corresponding element in \( A \):

• Top left: \( 5 - 4 = 1 \)
• Top right: \( -2 - 1 = -3 \)
• Bottom left: \( 1 - (-3) = 4 \)
• Bottom right: \( 3 - 2 = 1 \)

The result of \( A - B \) is then \( \begin{bmatrix} 1 & -3 \ 4 & 1 \end{bmatrix} \). This makes matrix subtraction a straightforward operation, as long as you carefully manage each element pairing.

Scalar Multiplication

In scalar multiplication, you multiply each element of a matrix by a constant number called a scalar. Unlike addition and subtraction, scalar multiplication doesn't require matrices to have the same dimensions because you are working with one matrix and one number.

For instance, let's multiply matrix \( A \) by the scalar 2:

• Matrix \( A = \begin{bmatrix} 5 & -2 \ 1 & 3 \end{bmatrix} \)
• Scalar: 2

To get \( 2A \), multiply each element in matrix \( A \) by 2:

• Top left: \( 2 \times 5 = 10 \)
• Top right: \( 2 \times (-2) = -4 \)
• Bottom left: \( 2 \times 1 = 2 \)
• Bottom right: \( 2 \times 3 = 6 \)

So, \( 2A = \begin{bmatrix} 10 & -4 \ 2 & 6 \end{bmatrix} \).

Scalar multiplication is useful for scaling the size of a matrix's elements without altering its structure.

Matrix Dimensions

Understanding matrix dimensions is a key part of performing matrix operations correctly. The dimensions of a matrix are defined by the number of its rows and columns, typically noted as \( m \times n \) where \( m \) is the number of rows and \( n \) is the number of columns.

For two matrices to be added or subtracted, they must have the same dimensions. This means their number of rows and columns must match exactly. For example, both matrices \( A \) and \( B \) in our exercise are \( 2 \times 2 \) matrices.

However, scalar multiplication doesn't require matrices to match dimensions. Instead, it involves using a single number, a scalar, to multiply every entry within the matrix.

Checking matrix dimensions is often the first step in any matrix operation. This ensures that the chosen operation can be performed without problems.