Problem 37 - Precalculus: Mathematics for Calculus

Question

The matrices \(A, B,\) and \(C\) are defined as follows. $$A=\left[\begin{array}{rrr}0.3 & 1.1 & 2.4 \\\0.9 & -0.1 & 0.4 \\\\-0.7 & 0.3 & -0.5\end{array}\right] \quad B=\left[\begin{array}{cc}1.2 & -0.1 \\\0 & -0.5 \\\0.5 & -2.1\end{array}\right]$$ $$C=\left[\begin{array}{rrr}-0.2 & 0.2 & 0.1 \\\1.1 & 2.1 & -2.1\end{array}\right]$$ Use a graphing calculator to carry out the indicated algebraic operation, or explain why it cannot be performed. $$A B$$

Step-by-Step Solution

Answer

Matrix multiplication is possible, and the result is \(\begin{bmatrix}1.56 & -5.62 \\ 1.28 & -0.88 \\ -1.09 & 0.97\end{bmatrix}\).

Step 1: Check Matrix Dimensions

To multiply matrices, the number of columns in the first matrix (Matrix A) must equal the number of rows in the second matrix (Matrix B). Matrix A is a 3x3 matrix, and Matrix B is a 3x2 matrix.

Step 2: Verify Multiplication Conditions

Matrix A has 3 columns, and Matrix B has 3 rows. Since these numbers are equal, the multiplication is possible.

Step 3: Perform Matrix Multiplication

The resultant matrix \(AB\) will have dimensions corresponding to the number of rows of Matrix A and the number of columns of Matrix B, resulting in a 3x2 matrix.

Step 4: Calculate Each Element of the Result Matrix

1. Calculate the element in the first row, first column: \((0.3 \times 1.2) + (1.1 \times 0) + (2.4 \times 0.5) = 0.36 + 0 + 1.2 = 1.56\).2. Calculate the element in the first row, second column: \((0.3 \times -0.1) + (1.1 \times -0.5) + (2.4 \times -2.1) = -0.03 - 0.55 - 5.04 = -5.62\).3. Calculate the element in the second row, first column: \((0.9 \times 1.2) + (-0.1 \times 0) + (0.4 \times 0.5) = 1.08 + 0 + 0.2 = 1.28\).4. Calculate the element in the second row, second column: \((0.9 \times -0.1) + (-0.1 \times -0.5) + (0.4 \times -2.1) = -0.09 + 0.05 - 0.84 = -0.88\).5. Calculate the element in the third row, first column: \((-0.7 \times 1.2) + (0.3 \times 0) + (-0.5 \times 0.5) = -0.84 + 0 - 0.25 = -1.09\).6. Calculate the element in the third row, second column: \((-0.7 \times -0.1) + (0.3 \times -0.5) + (-0.5 \times -2.1) = 0.07 - 0.15 + 1.05 = 0.97\).

Step 5: Write the Resultant Matrix

The final multiplied matrix \(AB\) is: \[AB = \begin{bmatrix}1.56 & -5.62 \1.28 & -0.88 \-1.09 & 0.97\end{bmatrix}\]

Key Concepts

Matrix Dimensions

Understanding the dimensions of matrices is crucial in matrix multiplication. A matrix is essentially a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix provide a compact way to describe its size by specifying the number of rows and the number of columns. For example:

• A 3x3 matrix, like Matrix A in this exercise, has three rows and three columns.
• A 3x2 matrix, like Matrix B, has three rows and two columns.

Knowing matrix dimensions helps determine if two matrices can be multiplied. To carry out the multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. This is a fundamental rule of matrix multiplication and ensures that each element of the resultant matrix can be calculated without ambiguity.

Resultant Matrix

When multiplying two matrices, the result is known as the resultant matrix. The dimensions of the resultant matrix are determined by the number of rows from the first matrix and the number of columns from the second matrix. For instance, when multiplying:

• Matrix A (3x3) by Matrix B (3x2),
• the resultant matrix will have dimensions of 3x2.

This means it will have three rows and two columns. During multiplication, each element of the resultant matrix is computed by taking the dot product of the corresponding row from the first matrix and the column from the second matrix. This involves multiplying elements from the row and column and then summing those products to get a single number, which fills a spot in the resultant matrix.

Multiplication Conditions

Matrix multiplication is only possible if specific conditions are met, primarily concerning the dimensions of the matrices. The primary condition is that the number of columns in the first matrix must match the number of rows in the second matrix. If this condition is met:

• You can proceed with the multiplication.
• Otherwise, the multiplication is undefined.

For example, in this exercise, Matrix A has 3 columns, and Matrix B has 3 rows, so the multiplication condition is satisfied. Notably, such conforming dimensions allow the element-wise multiplication and summation necessary for forming each element in the resultant matrix. This condition ensures that every element in the new matrix is the result of legitimate and complete operations.