Problem 16 - Precalculus

Question

Use the matrices \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] \quad B=\left[\begin{array}{rr}0 & -3 \\ -5 & 2\end{array}\right] \quad C=\left[\begin{array}{rrr}10 & -\frac{11}{2} & 0 \\ \frac{3}{5} & 5 & 9\end{array}\right]\) \(D=\left[\begin{array}{rr}7 & -13 \\ -\frac{4}{3} & 0 \\\ 6 & 8\end{array}\right] \quad E=\left[\begin{array}{rrr}1 & 2 & 3 \\ 0 & 4 & -9 \\ 0 & 0 & -5\end{array}\right]\) to compute the following or state that the indicated operation is undefined. $$ (A+B)(A-B) $$

Step-by-Step Solution

Answer

The result of \((A+B)(A-B)\) is \(\begin{bmatrix} -7 & 3 \\ 46 & 2 \end{bmatrix}\).

Step 1: Check Matrix Addition and Subtraction Compatibility

To perform the operations \((A+B)\) and \((A-B)\), the matrices \(A\) and \(B\) need to have the same dimensions. Both matrices \(A\) and \(B\) are 2x2 matrices, so we can proceed with the addition and subtraction.

Step 2: Compute Matrix A + B

Add corresponding elements of matrices \(A\) and \(B\): \[A+B = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} = \begin{bmatrix} 1+0 & 2+(-3) \ 3+(-5) & 4+2 \end{bmatrix} = \begin{bmatrix} 1 & -1 \ -2 & 6 \end{bmatrix}\]

Step 3: Compute Matrix A - B

Subtract corresponding elements of matrices \(B\) from \(A\): \[A-B = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} - \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} = \begin{bmatrix} 1-0 & 2-(-3) \ 3-(-5) & 4-2 \end{bmatrix} = \begin{bmatrix} 1 & 5 \ 8 & 2 \end{bmatrix}\]

Step 4: Multiply the Resulting Matrices (A + B)(A - B)

Calculate the product by using the formula for matrix multiplication, \([C]_{ij} = \sum_{k} A_{ik}B_{kj}\): \[(A+B)(A-B) = \begin{bmatrix} 1 & -1 \ -2 & 6 \end{bmatrix} \begin{bmatrix} 1 & 5 \ 8 & 2 \end{bmatrix} = \begin{bmatrix} (1)(1)+(-1)(8) & (1)(5)+(-1)(2) \ (-2)(1)+(6)(8) & (-2)(5)+(6)(2) \end{bmatrix} = \begin{bmatrix} 1 - 8 & 5 - 2 \ -2 + 48 & -10 + 12 \end{bmatrix} = \begin{bmatrix} -7 & 3 \ 46 & 2 \end{bmatrix}\]

Step 5: Finalize the Solution

The computation is complete. The result of \((A+B)(A-B)\) is \[\begin{bmatrix} -7 & 3 \ 46 & 2 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition is the process of adding two matrices by adding their corresponding elements. If you have matrices of the same dimensions—meaning they have the same number of rows and columns—you can proceed with their addition. For instance, consider matrices \(A\) and \(B\), which are both 2x2 matrices:

• Matrix \(A\): \( \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \)
• Matrix \(B\): \( \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} \)

To add these matrices, simply add each corresponding element:

• Top-left: \(1 + 0 = 1\)
• Top-right: \(2 + (-3) = -1\)
• Bottom-left: \(3 + (-5) = -2\)
• Bottom-right: \(4 + 2 = 6\)

Thus, \(A + B = \begin{bmatrix} 1 & -1 \ -2 & 6 \end{bmatrix}\).

Matrix Subtraction

Matrix subtraction is similar to matrix addition, but instead, we subtract the elements of one matrix from their corresponding elements in another matrix. Again, the matrices must have the same dimensions. Using the same matrices \(A\) and \(B\):

• Matrix \(A\): \( \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \)
• Matrix \(B\): \( \begin{bmatrix} 0 & -3 \ -5 & 2 \end{bmatrix} \)

Subtract each element of \(B\) from \(A\):

• Top-left: \(1 - 0 = 1\)
• Top-right: \(2 - (-3) = 5\)
• Bottom-left: \(3 - (-5) = 8\)
• Bottom-right: \(4 - 2 = 2\)

Thus, \(A - B = \begin{bmatrix} 1 & 5 \ 8 & 2 \end{bmatrix}\).

Matrix Multiplication

Matrix multiplication involves a more complex process than addition or subtraction. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. Here, we multiply the resulting matrices from our previous operations:

• \( A + B = \begin{bmatrix} 1 & -1 \ -2 & 6 \end{bmatrix} \)
• \( A - B = \begin{bmatrix} 1 & 5 \ 8 & 2 \end{bmatrix} \)

The product of these matrices is calculated by taking the dot product of rows from the first matrix with columns of the second:

• Top-left: \((1)(1) + (-1)(8) = 1 - 8 = -7\)
• Top-right: \((1)(5) + (-1)(2) = 5 - 2 = 3\)
• Bottom-left: \((-2)(1) + (6)(8) = -2 + 48 = 46\)
• Bottom-right: \((-2)(5) + (6)(2) = -10 + 12 = 2\)

So, the final product is \( \begin{bmatrix} -7 & 3 \ 46 & 2 \end{bmatrix} \).

2x2 Matrices

2x2 matrices are matrices that have two rows and two columns. They are one of the simplest forms of matrices and are often used in basic matrix operations due to their ease of computation. In these matrices, you only need to perform computations on four elements in each operation, such as addition, subtraction, or multiplication.
Each element in a 2x2 matrix plays a significant role. For instance, when performing matrix multiplication, every element from the rows of the first matrix is used with every element from the columns of the second matrix. Here's a basic structure of a 2x2 matrix:

• \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)

These four elements—\(a\), \(b\), \(c\), and \(d\)—are all you need to perform the basic operations described above, making 2x2 matrices a perfect starting point for learning matrix operations.

Matrix Dimensions

Understanding matrix dimensions is crucial in any matrix operation. The dimensions of a matrix are described by the number of rows and columns it has. For instance, a \(2x2\) matrix has two rows and two columns. This concept is vital because certain operations, like matrix addition and subtraction, require matrices to have the same dimensions.
When you see a matrix description like \(3x2\), it has 3 rows and 2 columns. Here's an example layout:

• \( \begin{bmatrix} a & b \ c & d \ e & f \end{bmatrix} \)

Remember, for two matrices to be added or subtracted, their dimensions must match exactly. But for multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Understanding these rules ensures you can perform matrix operations correctly and efficiently.