Problem 16 - Advanced Engineering Mathematics

Question

$$ \begin{aligned} &\text { If } \mathbf{A}=\left(\begin{array}{rr} -2 & 0 \\ 4 & 1 \\ 7 & 3 \end{array}\right) \text { and } \mathbf{B}=\left(\begin{array}{rr} 3 & -1 \\ 0 & 2 \\ -4 & -2 \end{array}\right), \text { find (a) } \mathbf{A}-\mathbf{B}\\\ &\text { (b) } \mathbf{B}-\mathbf{A},(\mathbf{c}) 2(\mathbf{A}+\mathbf{B}) . \end{aligned} $$

Step-by-Step Solution

Answer

(a) \( \begin{pmatrix} -5 & 1 \\ 4 & -1 \\ 11 & 5 \end{pmatrix} \), (b) \( \begin{pmatrix} 5 & -1 \\ -4 & 1 \\ -11 & -5 \end{pmatrix} \), (c) \( \begin{pmatrix} 2 & -2 \\ 8 & 6 \\ 6 & 2 \end{pmatrix} \).

Step 1: Set up matrices and operation for A-B

First, identify the matrices \( \mathbf{A} \) and \( \mathbf{B} \). We can perform matrix subtraction since both \( \mathbf{A} \) and \( \mathbf{B} \) have the same dimension (3x2). The operation \( \mathbf{A} - \mathbf{B} \) is done by subtracting each element of \( \mathbf{B} \) from the corresponding element of \( \mathbf{A} \).

Step 2: Perform matrix subtraction for A-B

Calculate \( \mathbf{A} - \mathbf{B} \):\[ \mathbf{A}-\mathbf{B} = \begin{pmatrix} -2 & 0 \ 4 & 1 \ 7 & 3 \end{pmatrix} - \begin{pmatrix} 3 & -1 \ 0 & 2 \ -4 & -2 \end{pmatrix} = \begin{pmatrix} -2-3 & 0-(-1) \ 4-0 & 1-2 \ 7-(-4) & 3-(-2) \end{pmatrix} \]\[ = \begin{pmatrix} -5 & 1 \ 4 & -1 \ 11 & 5 \end{pmatrix} \]

Step 3: Set up matrices and operation for B-A

Now, to find \( \mathbf{B} - \mathbf{A} \), we need to subtract each element of \( \mathbf{A} \) from the corresponding element of \( \mathbf{B} \).

Step 4: Perform matrix subtraction for B-A

Calculate \( \mathbf{B} - \mathbf{A} \):\[ \mathbf{B}-\mathbf{A} = \begin{pmatrix} 3 & -1 \ 0 & 2 \ -4 & -2 \end{pmatrix} - \begin{pmatrix} -2 & 0 \ 4 & 1 \ 7 & 3 \end{pmatrix} = \begin{pmatrix} 3-(-2) & -1-0 \ 0-4 & 2-1 \ -4-7 & -2-3 \end{pmatrix} \]\[ = \begin{pmatrix} 5 & -1 \ -4 & 1 \ -11 & -5 \end{pmatrix} \]

Step 5: Set up matrix addition for A+B

Next, calculate the sum \( \mathbf{A} + \mathbf{B} \). Add each element of \( \mathbf{A} \) to the corresponding element of \( \mathbf{B} \).

Step 6: Perform matrix addition for A+B

Calculate \( \mathbf{A} + \mathbf{B} \):\[ \mathbf{A}+\mathbf{B} = \begin{pmatrix} -2 & 0 \ 4 & 1 \ 7 & 3 \end{pmatrix} + \begin{pmatrix} 3 & -1 \ 0 & 2 \ -4 & -2 \end{pmatrix} = \begin{pmatrix} -2+3 & 0+(-1) \ 4+0 & 1+2 \ 7+(-4) & 3+(-2) \end{pmatrix} \]\[ = \begin{pmatrix} 1 & -1 \ 4 & 3 \ 3 & 1 \end{pmatrix} \]

Step 7: Multiply result of A+B by 2

Finally, multiply each element of the sum \( \mathbf{A} + \mathbf{B} \) by 2 for \( 2(\mathbf{A} + \mathbf{B}) \):\[ 2(\mathbf{A} + \mathbf{B}) = 2 \times \begin{pmatrix} 1 & -1 \ 4 & 3 \ 3 & 1 \end{pmatrix} = \begin{pmatrix} 2 \times 1 & 2 \times -1 \ 2 \times 4 & 2 \times 3 \ 2 \times 3 & 2 \times 1 \end{pmatrix} \]\[ = \begin{pmatrix} 2 & -2 \ 8 & 6 \ 6 & 2 \end{pmatrix} \]

Key Concepts

Matrix Subtraction

Matrix subtraction is a fundamental operation in the realm of matrix algebra. It is performed by subtracting each element of one matrix from the corresponding element of another matrix. Both matrices must have identical dimensions to ensure that each element has a counterpart to be subtracted from.
In this exercise, we performed subtraction between two matrices, \( \mathbf{A} \) and \( \mathbf{B} \).
To find \( \mathbf{A} - \mathbf{B} \), you follow these steps:

• Subtract the first element of \( \mathbf{B} \) from the first element of \( \mathbf{A} \).
• Continue this process for each corresponding element in both matrices.

For example, if you are given \( \mathbf{A} = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \) and \( \mathbf{B} = \begin{pmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{pmatrix} \), the result \( \mathbf{A} - \mathbf{B} = \begin{pmatrix} a_{11} - b_{11} & a_{12} - b_{12} \ a_{21} - b_{21} & a_{22} - b_{22} \end{pmatrix} \).
By performing these calculations on the given matrices, you end up with the subtracted matrix as shown in the solution.

Matrix Addition

Matrix addition is another essential matrix operation that involves adding corresponding elements of two matrices. Like subtraction, both matrices must share the same dimensions.
Adding matrices \( \mathbf{A} \) and \( \mathbf{B} \) involves the following steps:

• Add the first element of \( \mathbf{A} \) to the first element of \( \mathbf{B} \).
• Repeat this process for all corresponding elements.

For instance, given matrices \( \mathbf{A} = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \) and \( \mathbf{B} = \begin{pmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{pmatrix} \), the result of \( \mathbf{A} + \mathbf{B} = \begin{pmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{pmatrix} \).
This operation is commutative, meaning \( \mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A} \), and associative, which ensures that grouping of matrices does not affect the result.

Scalar Multiplication

Scalar multiplication is the process of multiplying each element of a matrix by a single number, which is called a scalar. This operation scales the entire matrix by this scalar and maintains the relative ratios between its elements.
In our example, once \( \mathbf{A} + \mathbf{B} \) has been calculated, we multiply every element by the scalar 2. The operations look like this:

• Multiply the first element of the matrix by 2.
• Apply this multiplication to each subsequent element in the matrix.

For a matrix \( \mathbf{C} = \begin{pmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{pmatrix} \), scalar multiplication by 2 yields \( 2 \mathbf{C} = \begin{pmatrix} 2c_{11} & 2c_{12} \ 2c_{21} & 2c_{22} \end{pmatrix} \).
This type of operation is very useful when scaling the outcomes of matrix equations or when expressing transformations in applied mathematics and physics.