Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: \(\left.A^{2}=A \cdot A\right)\) \(A=\left[\begin{array}{ll}1 & 0 \\ 2 & 3\end{array}\right], B=\left[\begin{array}{rrr}-2 & 3 & 4 \\ -1 & 1 & -5\end{array}\right], C=\left[\begin{array}{rr}0.5 & 0.1 \\ 1 & 0.2 \\ -0.5 & 0.3\end{array}\right], D=\left[\begin{array}{rrr}1 & 0 & -1 \\ -6 & 7 & 5 \\\ 4 & 2 & 1\end{array}\right]\) \(B D\)

Step-by-Step Solution

Answer

Multiplication possible; result is \(\begin{bmatrix} -4 & 29 & 21 \\ -27 & -3 & 1 \end{bmatrix}\).

Step 1: Check Dimensions for Multiplication Validity

To determine if the multiplication is possible, check the dimensions of matrices \(B\) and \(D\). Matrix \(B\) is a \(2 \times 3\) matrix, and matrix \(D\) is a \(3 \times 3\) matrix. Multiplication is possible if the number of columns in the first matrix equals the number of rows in the second matrix. Here, both conditions are satisfied (i.e., 3 columns in \(B\) to 3 rows in \(D\)). Thus, multiplication is possible.

Step 2: Perform Matrix Multiplication

Perform the matrix multiplication \(B \, D\). To find each entry of the result, multiply the elements of the rows of matrix \(B\) by the corresponding elements of the columns of matrix \(D\) and sum the products.\[B \cdot D = \begin{bmatrix} -2 & 3 & 4 \ -1 & 1 & -5 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & -1 \ -6 & 7 & 5 \ 4 & 2 & 1 \end{bmatrix} = \begin{bmatrix} -2 \cdot 1 + 3 \cdot -6 + 4 \cdot 4 & -2 \cdot 0 + 3 \cdot 7 + 4 \cdot 2 & -2 \cdot -1 + 3 \cdot 5 + 4 \cdot 1 \ -1 \cdot 1 + 1 \cdot -6 + (-5) \cdot 4 & -1 \cdot 0 + 1 \cdot 7 + (-5) \cdot 2 & -1 \cdot -1 + 1 \cdot 5 + (-5) \cdot 1 \end{bmatrix}\]Simplify each calculation:- First row, first column: \(-2 - 18 + 16 = -4\)- First row, second column: \(0 + 21 + 8 = 29\)- First row, third column: \(2 + 15 + 4 = 21\)- Second row, first column: \(-1 - 6 - 20 = -27\)- Second row, second column: \(0 + 7 - 10 = -3\)- Second row, third column: \(1 + 5 - 5 = 1\)

Step 3: Write Final Result

Construct the resultant \(2 \times 3\) matrix from the calculated values:\[\begin{bmatrix} -4 & 29 & 21 \ -27 & -3 & 1 \end{bmatrix}\]

Key Concepts

Matrix Dimensions

To understand matrix multiplication, it's crucial to first comprehend matrix dimensions. Each matrix is described by its size in terms of rows and columns. For instance, a matrix with 2 rows and 3 columns is termed a 2×3 matrix.
Understanding dimensions is vital because they dictate whether certain matrix operations are feasible. In matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. This alignment between rows and columns allows each element in a row of the first matrix to pair with a corresponding element in a column of the second matrix.
In our example, matrix B is 2×3, and matrix D is 3×3. Since B has 3 columns and D has 3 rows, multiplication is possible. Remember: matched dimensions make matrix multiplication viable!

Matrix Operations

Matrix operations include addition, subtraction, and multiplication, but multiplying matrices is distinctively crucial because it combines information differently than simple scalar multiplication.
Let's explore multiplication: when you multiply two matrices, you are essentially taking rows from the first matrix and columns from the second matrix. For each element of the resultant matrix, multiply the respective elements and sum those products to form each element of the new matrix.
Using matrices B and D as an example, imagine taking a row from B and a column from D. For each pair of elements within these row-column iterations, multiply together and then sum. This results in one number in the resultant matrix.

• For B's first row and D's first column, computations include: (-2×1) + (3×-6) + (4×4).
• Repeat this process for corresponding rows and columns to complete the new matrix.

Matrix multiplication isn't commutative (i.e., A×B ≠ B×A), which means the order of operations is crucial.

Resultant Matrix

After performing matrix multiplication, what you derive is the resultant matrix. With our given matrices B (2×3) and D (3×3), their product yields a matrix result following size rules—specifically, a 2×3 matrix.
The size of a resultant matrix is determined by the number of rows in the first matrix and the number of columns in the second matrix. For instance, the computed matrix in our exercise emerges from the operation \[B \cdot D = \begin{bmatrix} -4 & 29 & 21 \ -27 & -3 & 1 \end{bmatrix}\]These resulting elements capture the essence of the multiplication process—each element of the resultant matrix synthesizes information from row-column partnerships across matrices B and D.
This not only makes the resultant matrix a crucial component in linear algebra but also reflects the multi-dimensional nature of data and their relationships.