Problem 66 - A Graphical Approach to College Algebra

Question

Find each matrix product if possible. $$\left[\begin{array}{lll} 0 & 3 & -4 \end{array}\right]\left[\begin{array}{rrr} -2 & 6 & 3 \\ 0 & 4 & 2 \\ -1 & 1 & 4 \end{array}\right]$$

Step-by-Step Solution

Answer

The product of the matrices is \([4, 8, -10]\).

Step 1: Confirm Matrix Dimensions Compatibility

The first matrix is a 1x3 matrix, and the second matrix is a 3x3 matrix. To multiply matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Here, both numbers are 3, so the matrix multiplication is possible.

Step 2: Set Up the Resulting Matrix

Since the first matrix is 1x3 and the second matrix is 3x3, the resulting matrix will be 1x3. It will contain 3 elements.

Step 3: Compute Element in First Column

Multiply row elements of the first matrix with the corresponding column elements of the second matrix and sum them up: \[(0)(-2) + (3)(0) + (-4)(-1) = 0 + 0 + 4 = 4\]

Step 4: Compute Element in Second Column

Multiply row elements of the first matrix with the corresponding column elements of the second matrix and sum them up:\[(0)(6) + (3)(4) + (-4)(1) = 0 + 12 - 4 = 8\]

Step 5: Compute Element in Third Column

Multiply row elements of the first matrix with the corresponding column elements of the second matrix and sum them up: \[(0)(3) + (3)(2) + (-4)(4) = 0 + 6 - 16 = -10\]

Step 6: Construct Resulting Matrix

The resulting matrix from our calculations is: \[\left[\begin{array}{lll} 4 & 8 & -10 \end{array}\right]\]

Key Concepts

Matrix Dimensions

When discussing matrix multiplication, the first thing to understand is the dimensions of the matrices involved. A matrix is essentially a grid of numbers organized in rows and columns. Each matrix has dimensions denoted as "rows × columns". For successful multiplication, the number of columns in the first matrix must match the number of rows in the second matrix.

• In the exercise example, the first matrix is a 1x3 matrix, meaning it has 1 row and 3 columns.
• The second matrix is a 3x3 matrix, meaning it has 3 rows and 3 columns.

Since the first matrix has 3 columns and the second matrix has 3 rows, matrix multiplication is possible according to the rule.

Resulting Matrix

Understanding the size of the resulting matrix is key in the multiplication process. When two matrices are multiplied, the dimensions of the resulting matrix depend on the rows of the first matrix and the columns of the second matrix.

• The resulting matrix will have dimensions equal to the number of rows of the first matrix by the number of columns of the second matrix.
• In this example, the first matrix has 1 row and the second matrix has 3 columns.
• Thus, the resulting matrix will be a 1x3 matrix, meaning it will have one row with three elements.

This tells us what shape the solution of our matrix multiplication will take, crucial for organizing our results appropriately.

Element Computation

The computation of each element in the resulting matrix involves using the elements of the original matrices. Specifically, each element in the resulting matrix is obtained by the dot product of the corresponding row of the first matrix and the column of the second matrix.

Let's break it down from the example:

• First Element: We multiply and sum: \( (0)(-2) + (3)(0) + (-4)(-1) = 4 \)
• Second Element: We perform the same process: \( (0)(6) + (3)(4) + (-4)(1) = 8 \)
• Third Element: Similarly, we compute: \( (0)(3) + (3)(2) + (-4)(4) = -10 \)

Each element involves multiplying the corresponding components and summing them up, as seen above.
This ensures every computation in the resulting matrix is accurately calculated, leading to the final matrix: \[\left[\begin{array}{lll} 4 & 8 & -10 \end{array}\right]\]