Problem 74 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Multiply, if possible: \(\left[\begin{array}{rrr}4 & 2 & 0 \\ -1 & 3 & 1\end{array}\right]\left[\begin{array}{rr}0 & -2 \\ 3 & 1 \\ 5 & 0\end{array}\right]\)

Step-by-Step Solution

Answer

The product of the matrices is \( \begin{bmatrix} 6 & -6 \ 14 & 5 \end{bmatrix} \).

Step 1 - Verify Matrix Dimensions

Check the dimensions of the matrices to ensure that they can be multiplied. The first matrix has dimensions 2x3 and the second matrix has dimensions 3x2. Since the number of columns in the first matrix (3) matches the number of rows in the second matrix (3), they can be multiplied.

Step 2 - Write Down the Matrices

Explicitly write down the matrices for clear reference: \(\text{Matrix A} = \begin{bmatrix} 4 & 2 & 0 \ -1 & 3 & 1 \ \text{Matrix B} = \begin{bmatrix} 0 & -2 \ 3 & 1 \ 5 & 0 \end{bmatrix}\)

Step 3 - Multiply the Matrices

Perform the matrix multiplication by taking the dot product of the rows of the first matrix with the columns of the second matrix. For the element at position (1,1): \(4*0 + 2*3 + 0*5 = 6\)For the element at position (1,2): \(4*(-2) + 2*1 + 0*0 = -8 + 2 = -6\)For the element at position (2,1): \((-1)*0 + 3*3 + 1*5 = 9 + 5 = 14\)For the element at position (2,2): \((-1)*(-2) + 3*1 + 1*0 = 2 + 3 = 5\)

Step 4 - Construct the Resultant Matrix

Combine the results into a single matrix. The resulting matrix is: \(\begin{bmatrix} 6 & -6 \ 14 & 5 \end{bmatrix}\)

Key Concepts

Matrix Dimensions

Before performing matrix multiplication, it is essential to understand the dimensions of the matrices involved. Matrix dimensions are expressed as 'rows x columns'. For a multiplication to be valid, the number of columns in the first matrix must equal the number of rows in the second matrix. In this exercise, we have a 2x3 matrix and a 3x2 matrix. Since the number of columns in the first matrix (3) matches the number of rows in the second matrix (3), these matrices can be multiplied. You should always check this before starting any matrix multiplication.

Dot Product

The core operation in matrix multiplication is the dot product. To calculate a dot product, you multiply corresponding elements from a row of the first matrix by a column of the second matrix, and then sum these products.
For example, to find the element in the first row and first column of the resultant matrix, you take the first row of Matrix A and the first column of Matrix B: \( 4 * 0 + 2 * 3 + 0 * 5 = 6 \).
This process is repeated for each element in the resultant matrix. This operation ensures that each element of the new matrix is the sum of products, derived from the corresponding elements in the original matrices.

Resultant Matrix

Once all the dot products have been calculated, these values are assembled into the resultant matrix. The dimensions of the resultant matrix are determined by the number of rows in the first matrix and the number of columns in the second matrix.
In our example, Matrix A is 2x3 and Matrix B is 3x2, so the resultant matrix is 2x2.
After performing the necessary dot products, we combine the results: \( \begin{bmatrix} 6 & -6 \ 14 & 5 \end{bmatrix} \). This resultant matrix represents the final product of the multiplication process.

Step-by-Step Solution

Breaking down the matrix multiplication into simple steps can make the process clearer:

1. **Verify Matrix Dimensions**: Ensure the number of columns in the first matrix matches the number of rows in the second matrix.
2. **Document the Matrices**: Write down both matrices explicitly for easy reference.
3. **Multiply the Matrices**: Compute the dot product for each position in the resultant matrix.
4. **Construct the Resultant Matrix**: Assemble all computed values into the final matrix.
This step-by-step approach simplifies complex processes and helps avoid errors. Always follow these steps to ensure accurate and efficient matrix multiplication.