Problem 63 - A Graphical Approach to College Algebra

Question

Find each matrix product if possible. $$\left[\begin{array}{rrr} -2 & -3 & -4 \\ 2 & -1 & 0 \\ 4 & -2 & 3 \end{array}\right]\left[\begin{array}{rrr} 0 & 1 & 4 \\ 1 & 2 & -1 \\ 3 & 2 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

The product matrix is \([-15, -16, 3; -1, 0, 9; 7, 6, 12]\).

Step 1: Determine Matrix Compatibility

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The first matrix is a 3x3 matrix and the second matrix is also a 3x3 matrix, so they are compatible for multiplication.

Step 2: Multiply the First Row by Each Column of the Second Matrix

To find the elements of the first row of the resulting matrix, multiply the first row of the first matrix by each column of the second matrix: - Element (1,1) = i.e., (-2)*0 + (-3)*1 + (-4)*3 = 0 - 3 - 12 = -15. - Element (1,2) = i.e., (-2)*1 + (-3)*2 + (-4)*2 = -2 - 6 - 8 = -16. - Element (1,3) = i.e., (-2)*4 + (-3)*(-1) + (-4)*(-2) = -8 + 3 + 8 = 3.

Step 3: Multiply the Second Row by Each Column of the Second Matrix

Find the elements of the second row of the resulting matrix by multiplying the second row of the first matrix with each column of the second matrix: - Element (2,1) = i.e., 2*0 + (-1)*1 + 0*3 = 0 - 1 + 0 = -1. - Element (2,2) = i.e., 2*1 + (-1)*2 + 0*2 = 2 - 2 + 0 = 0. - Element (2,3) = i.e., 2*4 + (-1)*(-1) + 0*(-2) = 8 + 1 + 0 = 9.

Step 4: Multiply the Third Row by Each Column of the Second Matrix

To find the elements of the third row of the resulting matrix, multiply the third row of the first matrix by each column of the second matrix: - Element (3,1) = i.e., 4*0 + (-2)*1 + 3*3 = 0 - 2 + 9 = 7. - Element (3,2) = i.e., 4*1 + (-2)*2 + 3*2 = 4 - 4 + 6 = 6. - Element (3,3) = i.e., 4*4 + (-2)*(-1) + 3*(-2) = 16 + 2 - 6 = 12.

Step 5: Write the Resulting Matrix

Combine all the elements calculated to form the product matrix: \[\begin{array}{ccc}-15 & -16 & 3 \-1 & 0 & 9 \7 & 6 & 12 \end{array}\] This is the resultant matrix from the multiplication of the two given matrices.

Key Concepts

Matrix Compatibility

Matrix multiplication is a fundamental concept in mathematics that requires an understanding of matrix compatibility. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. This rule ensures that each element from the row is paired with an element from the column.

In our example, both matrices are 3x3, meaning the first matrix has 3 columns and the second matrix has 3 rows. This makes them compatible for multiplication. If the number of columns in the first matrix didn't match the rows in the second matrix, the product wouldn't be possible.

Think of compatibility as a handshake between matrices: each column meets a corresponding row to create a seamless connection. By respecting compatibility rules, you ensure the multiplication will yield a valid resultant matrix.

Matrix Elements Calculation

Calculating each element of the resulting matrix involves systematic multiplication and addition. When multiplying matrices, each element in the resulting matrix is found by taking a row from the first matrix and a column from the second matrix.

For example, to calculate the element in the first row, first column of the product matrix, you perform the operation:

• Multiply the elements of the first row of the first matrix by the corresponding elements in the first column of the second matrix.
• Add the results together.

For our example, element (1,1) is calculated as follows:

• o (-2) * 0 = 0
• o (-3) * 1 = -3
• o (-4) * 3 = -12
• o total = 0 - 3 - 12 = -15

This process is repeated for each element in the product matrix, with rows and columns progressively paired.

Understand that each calculation is a series of small multiplications and additions ensuring each new element is accurately constructed from existing data.

Resultant Matrix

After calculating all necessary elements, the final product is assembled into what is called the resultant matrix. This matrix represents the conclusion of multiplying two compatible matrices, and includes each element derived from earlier calculations.

In our exercise, the resultant matrix looks like this:

• element (1,1): -15
• element (1,2): -16
• element (1,3): 3
• element (2,1): -1
• element (2,2): 0
• element (2,3): 9
• element (3,1): 7
• element (3,2): 6
• element (3,3): 12

The resultant matrix is as follows:\[\begin{array}{ccc}-15 & -16 & 3 \-1 & 0 & 9 \7 & 6 & 12 \\end{array}\]
Understanding the structure and calculation of the resultant matrix is key to mastering matrix multiplication. Each entry is a piece of a larger puzzle that when assembled, gives the final complete product of the two matrices.