Question

Matrices \(A\) and \(B\) are given below. Simplify the given expression. $$ A=\left[\begin{array}{cc} 1 & -1 \\ 7 & 4 \end{array}\right] \quad B=\left[\begin{array}{cc} -3 & 2 \\ 5 & 9 \end{array}\right] $$ $$ 2(A-B)-(A-3 B) $$

Short answer

Result: \( \begin{bmatrix} -2 & 1 \\ 12 & 13 \end{bmatrix} \)

Step-by-step solution

Find Matrix A - B

First, subtract matrix B from matrix A. To do this, subtract corresponding elements of B from A: \[ A - B = \begin{bmatrix} 1 & -1 \ 7 & 4 \end{bmatrix} - \begin{bmatrix} -3 & 2 \ 5 & 9 \end{bmatrix} = \begin{bmatrix} 1 - (-3) & -1 - 2 \ 7 - 5 & 4 - 9 \end{bmatrix} = \begin{bmatrix} 4 & -3 \ 2 & -5 \end{bmatrix} \]

Multiply Matrix (A - B) by 2

Multiply each element of the newly found matrix (A - B) by 2: \[ 2(A - B) = 2 \begin{bmatrix} 4 & -3 \ 2 & -5 \end{bmatrix} = \begin{bmatrix} 2 \times 4 & 2 \times -3 \ 2 \times 2 & 2 \times -5 \end{bmatrix} = \begin{bmatrix} 8 & -6 \ 4 & -10 \end{bmatrix} \]

Find Matrix 3B

Next, multiply matrix B by 3: \[ 3B = 3 \begin{bmatrix} -3 & 2 \ 5 & 9 \end{bmatrix} = \begin{bmatrix} 3 \times -3 & 3 \times 2 \ 3 \times 5 & 3 \times 9 \end{bmatrix} = \begin{bmatrix} -9 & 6 \ 15 & 27 \end{bmatrix} \]

Find Matrix A - 3B

Subtract matrix 3B from A: \[ A - 3B = \begin{bmatrix} 1 & -1 \ 7 & 4 \end{bmatrix} - \begin{bmatrix} -9 & 6 \ 15 & 27 \end{bmatrix} = \begin{bmatrix} 1 - (-9) & -1 - 6 \ 7 - 15 & 4 - 27 \end{bmatrix} = \begin{bmatrix} 10 & -7 \ -8 & -23 \end{bmatrix} \]

Calculate Final Expression Result

Finally, subtract matrix (A - 3B) from matrix 2(A - B): \[ 2(A - B) - (A - 3B) = \begin{bmatrix} 8 & -6 \ 4 & -10 \end{bmatrix} - \begin{bmatrix} 10 & -7 \ -8 & -23 \end{bmatrix} = \begin{bmatrix} 8 - 10 & -6 - (-7) \ 4 - (-8) & -10 - (-23) \end{bmatrix} = \begin{bmatrix} -2 & 1 \ 12 & 13 \end{bmatrix} \]

Key concepts

Matrix Addition

Matrix addition is the process of adding two matrices by adding their corresponding components. For matrix addition to take place, the matrices should be of the same size, meaning they should have the same number of rows and columns.

Here's how matrix addition works,

• Consider two matrices \(C\) and \(D\) that have the same dimensions.
• To find \(C + D\), simply add each element in matrix \(C\) to the corresponding element in matrix \(D\).

For example, if \(C = \begin{bmatrix} 1 & 3 \ 5 & 7 \end{bmatrix}\) and \(D = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}\), then \(C + D = \begin{bmatrix} 1+2 & 3+4 \ 5+6 & 7+8 \end{bmatrix} = \begin{bmatrix} 3 & 7 \ 11 & 15 \end{bmatrix}\).

Matrix addition is both commutative and associative, which means:

• \(C + D = D + C\)
• \((C + D) + E = C + (D + E)\)

Matrix Subtraction

Matrix subtraction is very similar to matrix addition. Still, instead of adding, we subtract the corresponding elements of the matrices. Like addition, both matrices must also be of the same dimension.

To perform matrix subtraction,

• Given matrices \(E\) and \(F\) of the same dimensions, to find \(E-F\), subtract each element in matrix \(F\) from the corresponding element in matrix \(E\).

For instance, if \(E=\begin{bmatrix} 10 & 20 \ 30 & 40 \end{bmatrix}\) and \(F=\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\), then \(E-F=\begin{bmatrix} 10-1 & 20-2 \ 30-3 & 40-4 \end{bmatrix} = \begin{bmatrix} 9 & 18 \ 27 & 36 \end{bmatrix}\).

Matrix subtraction is not commutative, meaning \(E-F\) is not equal to \(F-E\). But it is still associative:

• \(E-(F-G) = E-F+G\)

Scalar Multiplication

Before diving into matrix expression simplification, let's understand scalar multiplication. Scalar multiplication involves multiplying every entry in a matrix by a scalar value (a single number).

Here's how you perform scalar multiplication,

• Consider a matrix \(G\) and a scalar \(s\).
• To find \(sG\), multiply every entry in matrix \(G\) by the scalar \(s\).

For example, if \(G = \begin{bmatrix} 3 & 6 \ 9 & 12 \end{bmatrix}\) and \(s = 2\), then \(2G = \begin{bmatrix} 2 \times 3 & 2 \times 6 \ 2 \times 9 & 2 \times 12 \end{bmatrix} = \begin{bmatrix} 6 & 12 \ 18 & 24 \end{bmatrix}\).

Scalar multiplication is a crucial tool in simplifying expressions involving matrices. It is also commutative with matrix addition and subtraction, allowing us to rearrange expressions to simplify as needed.

Matrix Expression Simplification

Simplifying matrix expressions often involves several operations, such as matrix addition, subtraction, and scalar multiplication you have learned earlier. The goal is to reduce a complex expression into a simpler, equivalent form, ultimately finding an answer in its simplest representation.

Follow these general steps to simplify matrix expressions:

• Break down complex expressions into simpler parts using parentheses and follow the correct sequence of operations based on the order of operations.
• Apply matrix addition or subtraction to combine like terms or simplify differences.
• Use scalar multiplication to adjust the matrices before any addition or subtraction, especially when dealing with factors involved around matrices.

With the expression \(2(A-B)-(A-3B)\), start by simplifying each component:

• First, find \(A-B\) by subtracting each element in \(B\) from the corresponding one in \(A\).
• Then, apply scalar multiplication to \(A-B\) by multiplying with 2, giving \(2(A-B)\).
• Next, compute \(3B\) by multiplying each element in \(B\) by 3. Subtract \(3B\) from \(A\) to get \(A-3B\).
• Lastly, find the answer by subtracting \(A-3B\) from \(2(A-B)\).

This simplified form represents the solution of the original matrix expression.