Problem 40 - 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. $$ A=\left[\begin{array}{rr} 4 & 0 \\ -3 & 5 \\ 0 & 1 \end{array}\right] \quad B=\left[\begin{array}{rr} 5 & 1 \\ -2 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right] $$ $$ A(B+C) $$

Step-by-Step Solution

Answer

The resultant matrix of A(B+C) is \[ \left[\begin{array}{cc} 24 & 0 \ -33 & -5 \ -3 & -1 \end{array}\right] \]

Step 1: Add matrices B and C

Matrix B and C are both \(2x2\) matrices. They add together by adding the respective elements in their corresponding positions. \[ B+C = \left[\begin{array}{cc} 5+1 & 1+(-1) \ -2+(-1) & -2+1 \end{array}\right] = \left[\begin{array}{cc} 6 & 0 \ -3 & -1 \end{array}\right] \]

Step 2: Multiply matrix A with (B+C)

Matrix multiplication involves taking each row from the first matrix and multiplying it by each column in the second matrix, adding the results to create the new matrix. \[ A(B+C) = \left[\begin{array}{cc} 4 & 0 \ -3 & 5 \ 0 & 1 \end{array}\right] \times \left[\begin{array}{cc} 6 & 0 \ -3 & -1 \end{array}\right] = \left[\begin{array}{cc} 4*6+0*(-3) & 4*0+0*(-1) \ -3*6+5*(-3) & -3*0+5*(-1) \ 0*6+1*(-3) & 0*0+1*(-1) \end{array}\right] = \left[\begin{array}{cc} 24 & 0 \ -33 & -5 \ -3 & -1 \end{array}\right] \]

Key Concepts

Matrix Addition

In the context of algebra, matrix addition is a fundamental operation where two matrices of the same dimensions can be added together to create a new matrix. For the addition to be valid, each matrix must have the same number of rows and columns. The resulting matrix is derived by simply adding corresponding elements from each matrix.

Let's consider an example using matrices B and C from the exercise provided. Both B and C are 2x2 matrices. To perform addition, we align them element by element, and add the numbers that are in the same position:

• The entry in the first row, first column of matrix B (which is 5) is added to the entry in the first row, first column of matrix C (which is 1), yielding 6.
• This process is repeated for each element, always combining the numbers at matching positions.

The rule is simple: only matrices with matching dimensions can be added. If the matrices have different dimensions, the operation cannot be performed and is considered 'undefined'.

Matrix Multiplication

Matrix multiplication, unlike addition, involves a more complex procedure and cannot be carried out between any two matrices. The number of columns in the first matrix must be equal to the number of rows in the second matrix for the multiplication to be possible. The resulting matrix will have the number of rows of the first matrix and the number of columns of the second one.

To multiply two matrices, we take the 'dot product' of rows and columns: each entry in the resulting matrix is the sum of the products of corresponding elements from the row of the first matrix and the column of the second matrix.

The Steps in Matrix Multiplication:

• Select the row from the first matrix.
• Select the column from the second matrix.
• Multiply each element from the row by the corresponding element from the column, then sum those products.
• Place the resulting sum in the appropriate position in the new matrix.

This operation requires a bit of practice to perform quickly, but understanding the fundamental procedure is essential. In our exercise, since matrix A is a 3x2 matrix and the result of B+C is a 2x2 matrix, we can proceed with the multiplication, which results in a 3x2 matrix.

Matrix Dimensions

Matrix dimensions are crucial in determining whether certain operations can be performed. The dimensions of a matrix refer to the number of rows and columns it contains, usually noted as 'rows x columns'.

Understanding the dimensions is not only important for matrix addition, as we've seen, but it is vital when it comes to matrix multiplication. For example, if we have a 3x2 matrix (3 rows and 2 columns) and we wish to multiply it by another matrix, the second matrix must have 2 rows and an arbitrary number of columns. The resulting matrix from this multiplication will then have dimensions that match the number of rows of the first matrix and the number of columns of the second.

Whenever facing matrix operations, always start by verifying dimensions to ensure that the operation is defined. For instance, adding a 3x2 matrix to a 2x2 matrix isn't possible because the dimensions don't match. Similarly, multiplying a 3x2 matrix by a 3x3 is not possible because the number of columns in the first does not match the number of rows in the second.