Problem 20 - College Algebra

Question

Use the matrices below to perform matrix multiplication. \(A=\left[\begin{array}{rr}-1 & 5 \\ 3 & 2\end{array}\right], B=\left[\begin{array}{rrr}3 & 6 & 4 \\ -8 & 0 & 12\end{array}\right], C=\left[\begin{array}{rr}4 & 10 \\ -2 & 6 \\ 5 & 9\end{array}\right], D=\left[\begin{array}{rrr}2 & -3 & 12 \\ 9 & 3 & 1 \\ 0 & 8 & -10\end{array}\right]\) \(\mathrm{CA}\)

Step-by-Step Solution

Answer

The matrix \(CA\) is \(\begin{bmatrix} 26 & 40 \\ 20 & 2 \\ 22 & 43 \end{bmatrix}\).

Step 1: Understand Matrix Multiplication Compatibility

To multiply two matrices, say \(X\) of size \(m \times n\) and \(Y\) of size \(n \times p\), the number of columns in \(X\) (\(n\)) must be equal to the number of rows in \(Y\) (\(n\)). Here, matrix \(C\) is of size \(3 \times 2\) and matrix \(A\) is of size \(2 \times 2\). Since the number of columns in \(C\) matches the number of rows in \(A\), matrix multiplication is possible. The resulting matrix will be \(3 \times 2\).

Step 2: Set Up the Multiplication

The resulting matrix, \(CA\), will be of size \(3 \times 2\). Each element \((i,j)\) of matrix \(CA\) will be computed as the dot product of the \(i^{th}\) row of \(C\) and the \(j^{th}\) column of \(A\). Here, we will calculate each element step-by-step.

Step 3: Calculate Element (1,1) of CA

Compute element \((1,1)\) by taking the dot product of the first row of \(C\) and the first column of \(A\): \[4 \times (-1) + 10 \times 3 = -4 + 30 = 26\].

Step 4: Calculate Element (1,2) of CA

Compute element \((1,2)\) by taking the dot product of the first row of \(C\) and the second column of \(A\): \[4 \times 5 + 10 \times 2 = 20 + 20 = 40\].

Step 5: Calculate Element (2,1) of CA

Compute element \((2,1)\) by taking the dot product of the second row of \(C\) and the first column of \(A\): \[-2 \times (-1) + 6 \times 3 = 2 + 18 = 20\].

Step 6: Calculate Element (2,2) of CA

Compute element \((2,2)\) by taking the dot product of the second row of \(C\) and the second column of \(A\): \[-2 \times 5 + 6 \times 2 = -10 + 12 = 2\].

Step 7: Calculate Element (3,1) of CA

Compute element \((3,1)\) by taking the dot product of the third row of \(C\) and the first column of \(A\): \[5 \times (-1) + 9 \times 3 = -5 + 27 = 22\].

Step 8: Calculate Element (3,2) of CA

Compute element \((3,2)\) by taking the dot product of the third row of \(C\) and the second column of \(A\): \[5 \times 5 + 9 \times 2 = 25 + 18 = 43\].

Step 9: Assemble the Resulting Matrix CA

The resulting matrix \(CA\) is composed of the computed elements: \[CA = \begin{bmatrix} 26 & 40 \ 20 & 2 \ 22 & 43 \end{bmatrix}\].

Key Concepts

Dot Product

The dot product is a crucial part of matrix multiplication. It involves multiplying corresponding elements of a row and a column, and then adding these products. For instance, when calculating an element of a resulting matrix, you take a specific row from the first matrix and a corresponding column from the second matrix. Each pair of elements in this row and column are multiplied together, and the results are summed.

For example, to compute an element \(ij\) in a matrix formed by multiplying two given matrices, it's written as: \[ c_{ij} = a_{i1} \cdot b_{1j} + a_{i2} \cdot b_{2j} + \, ... + a_{in} \cdot b_{nj} \]

This process totals each element and places the sum in new matrix which holds position (i,j). Understanding this helps in performing matrix multiplications smoothly.

Matrix Compatibility

Matrix multiplication isn't always possible; this brings us to the concept of matrix compatibility. Two matrices can be multiplied only if the number of columns in the first matrix is equal to the number of rows in the second matrix.

If you have a matrix \(X\) of dimensions \(m \times n\) and another matrix \(Y\) of dimensions \(n \times p\), then you can multiply them to get a resulting matrix. In our case, matrix \(C\) (size \(3 \times 2\)) and matrix \(A\) (size \(2 \times 2\)) are compatible, since C has 2 columns and A has 2 rows.

This is a crucial rule to remember whenever you're checking to see if matrix multiplication is possible.

Matrix Dimensions

Every matrix is defined by its dimensions, which specify the number of rows and columns it contains. When performing operations like multiplication, understanding matrix dimensions is vital.

For example, a matrix \(C\) with dimensions \(3 \times 2\) has 3 rows and 2 columns, whereas matrix \(A\) with dimensions \(2 \times 2\) has 2 rows and 2 columns. These dimensions help you to determine if two matrices can be multiplied and also what the dimensions of the resulting matrix will be.

Always start by clearly identifying and marking the dimensions, as it precludes errors of compatibility that could arise later.

Resulting Matrix

After you've confirmed compatibility and carried out the necessary dot products, the outcome is known as the resulting matrix. This new matrix has its own dimensions determined by the number of rows from the first matrix and the number of columns from the second.

Using matrices \(C\) and \(A\) as an example, the resulting matrix from their product \(CA\) will have a dimension of \(3 \times 2\), as C contributes the rows and A contributes the columns.

This is important as it guides you in knowing what size your answer should be and ensures your computations adhere to expected formats. The process invariably leads to a structured and accurate result.