Problem 4 - Precalculus

Question

For each pair of matrices \(A\) and \(B\) find the following, if defined \- \(3 A\) \- \(-B\) \- \(A^{2}\) \- \(A-2 B\) \- \(A B\) \- \(B A\) $$ A=\left[\begin{array}{ll} 2 & 4 \\ 6 & 8 \end{array}\right], B=\left[\begin{array}{rrr} -1 & 3 & -5 \\ 7 & -9 & 11 \end{array}\right] $$

Step-by-Step Solution

Answer

\(3A = \begin{bmatrix} 6 & 12 \\ 18 & 24 \end{bmatrix}\), \(-B = \begin{bmatrix} 1 & -3 & 5 \\ -7 & 9 & -11 \end{bmatrix}\), \(A^2 = \begin{bmatrix} 28 & 40 \\ 60 & 88 \end{bmatrix}\), \(A - 2B\) and \(BA\) are undefined, \(AB = \begin{bmatrix} 26 & -30 & 34 \\ 50 & -54 & 58 \end{bmatrix}\).

Step 1: Compute 3A

To find \(3A\), multiply each element of matrix \(A\) by 3.\[3A = 3 \times \begin{bmatrix}2 & 4 \6 & 8 \end{bmatrix} = \begin{bmatrix}6 & 12 \18 & 24\end{bmatrix}\]

Step 2: Compute -B

To find \(-B\), multiply each element of matrix \(B\) by -1.\[-B = -1 \times \begin{bmatrix}-1 & 3 & -5 \7 & -9 & 11\end{bmatrix} = \begin{bmatrix}1 & -3 & 5 \-7 & 9 & -11\end{bmatrix}\]

Step 3: Compute A²

Matrix \(A^2\) is \(A\) multiplied by itself. Multiply matrix \(A\) by matrix \(A\):\[A^2 = \begin{bmatrix}2 & 4 \6 & 8\end{bmatrix}\begin{bmatrix}2 & 4 \6 & 8\end{bmatrix} = \begin{bmatrix}2 \times 2 + 4 \times 6 & 2 \times 4 + 4 \times 8 \6 \times 2 + 8 \times 6 & 6 \times 4 + 8 \times 8\end{bmatrix} = \begin{bmatrix}28 & 40 \60 & 88\end{bmatrix}\]

Step 4: Compute A - 2B

To find \(A - 2B\), first compute \(2B\), and then subtract it from \(A\):\[2B = 2 \times \begin{bmatrix}-1 & 3 & -5 \7 & -9 & 11\end{bmatrix} = \begin{bmatrix}-2 & 6 & -10 \14 & -18 & 22\end{bmatrix}\]Since \(A\) and \(2B\) are not of the same size (\(A\) is 2x2, \(2B\) is 2x3), \(A - 2B\) is undefined.

Step 5: Compute AB

To find \(AB\), multiply matrix \(A\) by matrix \(B\):\[AB = \begin{bmatrix}2 & 4 \6 & 8\end{bmatrix}\begin{bmatrix}-1 & 3 & -5 \7 & -9 & 11\end{bmatrix} = \begin{bmatrix}2 \times (-1) + 4 \times 7 & 2 \times 3 + 4 \times (-9) & 2 \times (-5) + 4 \times 11 \6 \times (-1) + 8 \times 7 & 6 \times 3 + 8 \times (-9) & 6 \times (-5) + 8 \times 11\end{bmatrix} = \begin{bmatrix}26 & -30 & 34 \50 & -54 & 58\end{bmatrix}\]

Step 6: Compute BA

To find \(BA\), multiply matrix \(B\) by matrix \(A\):\[BA = \begin{bmatrix}-1 & 3 & -5 \7 & -9 & 11\end{bmatrix}\begin{bmatrix}2 & 4 \6 & 8 \end{bmatrix}\]Since the number of columns in \(B\) (which is 3) does not match the number of rows in \(A\) (which is 2), \(BA\) is undefined.

Key Concepts

Matrix Operations

Matrix operations involve a variety of techniques that allow you to manipulate matrices. These techniques include addition, subtraction, multiplication, and even powers of matrices. Each operation follows specific rules due to the nature of matrices.

• **Addition and Subtraction:** Only matrices of the same size can be added or subtracted. You perform these operations by adding or subtracting corresponding elements.
• **Multiplication:** Matrix multiplication is more complex and involves multiplying rows by columns. It's vital to remember that the number of columns in the first matrix must match the number of rows in the second matrix for multiplication to be defined.

Matrix operations are foundational in linear algebra and applicable in various fields such as computer graphics and physics.

Scalar Multiplication

Scalar multiplication is the process of multiplying each element of a matrix by a scalar value, which is simply a regular number. This operation scales the entire matrix, enlarging or reducing it depending on the scalar.
For example, given matrix \(A\):\[A = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}\]Multiplying by a scalar, say 3, looks like this:\[3A = 3 \times \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 12 \ 18 & 24 \end{bmatrix}\]This simple process is an easy way to handle transformations and scaling in geometric contexts, as well as in solving linear equations.

Matrix Square

Squaring a matrix involves multiplying the matrix by itself. Remember, this is only possible when the matrix is a square matrix, meaning it has the same number of rows and columns.
Consider the previous matrix \(A\):\[A = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}\]To compute \(A^2\), multiply \(A\) by itself:\[A^2 = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix} \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix} = \begin{bmatrix} 28 & 40 \ 60 & 88 \end{bmatrix}\]This operation is common in applied mathematics, including solving systems of linear equations and modeling in physics.

Matrix Addition and Subtraction

Matrix addition and subtraction require that matrices have the same dimensions. You add or subtract corresponding elements to complete the operation.
Given matrices \(A\) and \(B\), addition is straightforward, provided both matrices are of the same size. However, in cases like subtracting \(2B \) from \( A \), where \(A\) (2x2) and \(B\) (2x3) are of different sizes, direct subtraction is impossible due to dimension mismatch.
Steps to ensure successful addition or subtraction include:

• Check matrix dimensions first.
• Align corresponding elements before performing operations.

This understanding is crucial for efficiency in more advanced algebra and data manipulation tasks.