Problem 39 - College Algebra

Question

In Exercises \(37-44\), perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows If an operation is not defined, state the reason. $$A=\left[\begin{array}{rr}4 & 0 \\\\-3 & 5 \\\0 & 1\end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\\\-2 & -2\end{array}\right] \quad C=\left[\begin{array}{rr}1 & -1 \\\\-1 & 1\end{array}\right]$$ $$B C+C B$$

Step-by-Step Solution

Answer

The result of the matrix operation \(B C+C B\) is \( \begin{bmatrix}11 & -1\\-3 & 7\end{bmatrix}\)

Step 1: Checking the Compatibility of Matrices

Firstly, check whether the operations on the matrices are defined or not. The multiplication of two matrices is only defined if the number of columns of the first matrix is equal to the number of rows of the second matrix. As both matrices B and C are \(2x2\), they can be multiplied.

Step 2: Multiply Matrices B and C

Let's calculate \(BC\). The element in the \(i^{th}\) row and \(j^{th}\) column of the result is obtained by multiplying every element of the \(i^{th}\) row of the first matrix by the corresponding element of the \(j^{th}\} column of the second matrix and adding them up. So, \(BC= \begin{bmatrix}5*1+1*-1 & 5*-1+1*1\\-2*1+(-2)*-1 & -2*-1+(-2)*1\end{bmatrix} = \begin{bmatrix}4 & -4\\0 & 4\end{bmatrix}\)

Step 3: Multiply Matrices C and B

Then, calculate \(CB\). Apply the same rule of matrix multiplication like previous step to get the product. Thus, \(CB=\begin{bmatrix}1*5+(-1)*(-2) & 1*1+(-1)*-2\\-1*5+1*(-2) & -1*1+1*2 \end{bmatrix}= \begin{bmatrix}7 & 3\\-3 & 3\end{bmatrix}\)

Step 4: Add the Resultant Matrices

Finally, add up the two resultant matrices (\(BC\) and \(CB\)) from previous steps. So, \(BC + CB= \begin{bmatrix}4 & -4\\0 & 4\end{bmatrix} + \begin{bmatrix}7 & 3\\-3 & 3\end{bmatrix}= \begin{bmatrix}4+7 & -4+3\\0+(-3) & 4+3\end{bmatrix}= \begin{bmatrix}11 & -1\\-3 & 7\end{bmatrix} \)

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential concept in linear algebra, used to combine two matrices. Unlike addition, the order in multiplication matters. Here's how to perform it step by step:

• Check Dimensions: To multiply Matrix A by Matrix B, the number of columns in A must equal the number of rows in B.
• Multiply and Add: The element at the position \(i, j\) in the product matrix is obtained by multiplying the elements of the \(i\)th row of the first matrix with the \(j\)th column of the second matrix, then summing them up.

For example, given matrices B and C both 2x2, you can multiply them because their inner dimensions match. Calculations showed that:\[BC= \begin{bmatrix}4 & -4\0 & 4\end{bmatrix}\]Matrix multiplication is not commutative, meaning \(BC eq CB\). It’s vital to carefully follow these steps for each specific order of multiplication.

Matrix Addition

Matrix addition is useful for combining two matrices of the same dimensions. Unlike multiplication, addition is simple and straightforward:

• Same Dimensions: Ensure both matrices have the same number of rows and columns.
• Add Corresponding Elements: Add each element in one matrix to the corresponding element in the other matrix.

For the given exercise, matrices \(BC\) and \(CB\) resulted in:\[BC = \begin{bmatrix}4 & -4\0 & 4\end{bmatrix}, \quad CB = \begin{bmatrix}7 & 3\-3 & 3\end{bmatrix}\]Add these element by element to produce:\[BC + CB = \begin{bmatrix}11 & -1\-3 & 7\end{bmatrix}\]This straightforward method makes matrix addition a clear and valuable operation.

Compatibility of Matrices

Understanding compatibility is key to efficient matrix operations. Here’s what you should consider:

• For Multiplication: The number of columns in the first matrix must match the number of rows in the second matrix.
• For Addition: Both matrices must have identical dimensions, meaning the same number of rows and columns.

In our example with matrices B and C, they were both 2x2, meeting the criteria for both multiplication and addition. Checking compatibility ensures operations are defined and performed correctly. If matrices do not meet these criteria, the attempted operation will result in an error or be undefined. Always verify dimensions first before proceeding with any matrix operations.