Problem 46 - College Algebra

Question

Let $$\begin{aligned}&A=\left[\begin{array}{ll}1 & 0 \\\0 & 1\end{array}\right], \quad B=\left[\begin{array}{rr}1 & 0 \\\0 & -1\end{array}\right], \quad C=\left[\begin{array}{rr}-1 & 0 \\ 0 & 1\end{array}\right]\\\&D=\left[\begin{array}{rr}-1 & 0 \\\0 & -1\end{array}\right]\end{aligned}$$ Find the product of the difference between \(A\) and \(B\) and the sum of \(C\) and \(D\)

Step-by-Step Solution

Answer

The product of the difference between A and B, and the sum of C and D is the matrix \(\left[\begin{array}{cc}0 & 0 \ 0 & 4\end{array}\right]\)

Step 1: Find the Difference Between A and B

Start by finding the difference between matrices A and B. The difference of two matrices is found by subtracting corresponding entries from each matrix. So, we have:\n\[ A - B = \left[\begin{array}{cc}1-1 & 0-0 \ 0-0 & 1-(-1)\end{array}\right] = \left[\begin{array}{cc}0 & 0 \ 0 & 2\end{array}\right]\]

Step 2: Find the Sum of C and D

Next, find the sum of matrices C and D. The sum of two matrices is found by adding corresponding entries from each matrix. So, we have: \n\[ C + D = \left[\begin{array}{cc}-1-(-1) & 0-0 \ 0-0 & 1-(-1)\end{array}\right] = \left[\begin{array}{cc}0 & 0 \ 0 & 2\end{array}\right]\]

Step 3: Compute the Product of the Resultant Matrices

Finally, compute the product of the two resultant matrices from steps 1 and 2. To do this, we multiply the first row of the first matrix with the first column of the second matrix to get the entry of the first row and first column of the product matrix. We then multiply the second row of the first matrix with the second column of the second matrix to get the entry of the second row and second column of the product matrix. This gives:\n\[ (A-B) * (C+D) = \left[\begin{array}{cc}0 & 0 \ 0 & 2\end{array}\right] * \left[\begin{array}{cc}0 & 0 \ 0 & 2\end{array}\right] = \left[\begin{array}{cc}0*0 + 0*0 & 0*0+0*2 \ 0*0 + 2*0 & 0*0 + 2*2\end{array}\right] = \left[\begin{array}{cc}0 & 0 \ 0 & 4\end{array}\right]\]

Key Concepts

Matrix Addition

Matrix addition is one of the fundamental operations we perform on matrices. It involves adding two matrices by adding their corresponding elements. Both matrices should have the same dimension for this operation to be valid.
For example, if you have two matrices, both being 2x2 in size, matrix addition is performed as follows:

• Take the element in the first row, first column of the first matrix and add it to the element in the first row, first column of the second matrix.
• Repeat the same operation for each corresponding element in the matrices.

Given the matrices C and D:\[C = \begin{bmatrix} -1 & 0 \ 0 & 1 \end{bmatrix}, D = \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix}\]the sum becomes:\[C + D = \begin{bmatrix} -1+(-1) & 0+0 \ 0+0 & 1+(-1) \end{bmatrix} = \begin{bmatrix} -2 & 0 \ 0 & 0 \end{bmatrix}\]

Matrix Subtraction

Matrix subtraction is similar to matrix addition in that it also requires matrices of the same size. Instead of adding elements from each matrix, you will subtract the corresponding elements of the second matrix from the first. Just follow these steps:

• Take each element from the first matrix and subtract the corresponding element in the second matrix.
• Do this for each position in the matrix grid.

To subtract matrices A and B:\[A = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}, B = \begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix}\]the difference is:\[A - B = \begin{bmatrix} 1-1 & 0-0 \ 0-0 & 1-(-1) \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 2 \end{bmatrix}\] This simple subtraction reveals how elements interact across matrices in operations.

Matrix Multiplication

Matrix multiplication is often more complex than addition or subtraction. In matrix multiplication, the number of columns in the first matrix should match the number of rows in the second matrix. For two 2x2 matrices the process is as follows:

• Multiply the elements of the rows of the first matrix by the elements of the columns in the second matrix.
• Sum the products of these multiplications to get each element of the resulting matrix.

Using the matrices that we derived from earlier steps:\[(A-B) = \begin{bmatrix} 0 & 0 \ 0 & 2 \end{bmatrix}, (C+D) = \begin{bmatrix} 0 & 0 \ 0 & 2 \end{bmatrix}\]The multiplication of these matrices is:\[(A-B) \times (C+D) = \begin{bmatrix} 0*0 + 0*0 & 0*0 + 0*2 \ 0*0 + 2*0 & 0*0 + 2*2 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 4 \end{bmatrix}\] Notice how the 2's interact to form 4 in the final product. This process highlights how multiplication can compound the effects seen in addition and subtraction of matrices.