Problem 49 - 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: \(\left.A^{2}=A \cdot A\right)\) \(A=\left[\begin{array}{ll}1 & 0 \\ 2 & 3\end{array}\right], B=\left[\begin{array}{rrr}-2 & 3 & 4 \\ -1 & 1 & -5\end{array}\right], C=\left[\begin{array}{rr}0.5 & 0.1 \\ 1 & 0.2 \\ -0.5 & 0.3\end{array}\right], D=\left[\begin{array}{rrr}1 & 0 & -1 \\ -6 & 7 & 5 \\\ 4 & 2 & 1\end{array}\right]\) \(A(B C)\)

Step-by-Step Solution

Answer

The product \( A(BC) \) is \( \begin{bmatrix} 0 & 1.6 \\ 9 & -1 \end{bmatrix}. \)

Step 1: Understand Matrix Multiplication Dimensional Requirements

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Given matrix dimensions: \( A \) is a 2x2 matrix, \( B \) is a 2x3 matrix, and \( C \) is a 3x2 matrix.

Step 2: Compute Product BC

Matrix \( B \) (2x3) can be multiplied by matrix \( C \) (3x2) since the number of columns in \( B \) matches the number of rows in \( C \). Perform matrix multiplication to find \( BC \): \[BC = \begin{bmatrix} -2 & 3 & 4 \ -1 & 1 & -5 \end{bmatrix} \times \begin{bmatrix} 0.5 & 0.1 \ 1 & 0.2 \ -0.5 & 0.3 \end{bmatrix}\]Calculate each element of the resulting 2x2 matrix:- First row, first column: \((-2)(0.5) + (3)(1) + (4)(-0.5) = -1 + 3 - 2 = 0\)- First row, second column: \((-2)(0.1) + (3)(0.2) + (4)(0.3) = -0.2 + 0.6 + 1.2 = 1.6\)- Second row, first column: \((-1)(0.5) + (1)(1) + (-5)(-0.5) = -0.5 + 1 + 2.5 = 3\)- Second row, second column: \((-1)(0.1) + (1)(0.2) + (-5)(0.3) = -0.1 + 0.2 - 1.5 = -1.4\)So, \[ BC = \begin{bmatrix} 0 & 1.6 \ 3 & -1.4 \end{bmatrix} \].

Step 3: Compute Product A(BC)

Matrix \( A \) (2x2) can be multiplied by \( BC \) (2x2) since their dimensions are compatible for multiplication. Use matrix multiplication to find \( A(BC) \):\[A(BC) = \begin{bmatrix} 1 & 0 \ 2 & 3 \end{bmatrix} \times \begin{bmatrix} 0 & 1.6 \ 3 & -1.4 \end{bmatrix}\]Calculate each element of the resulting matrix:- First row, first column: \((1)(0) + (0)(3) = 0\)- First row, second column: \((1)(1.6) + (0)(-1.4) = 1.6\)- Second row, first column: \((2)(0) + (3)(3) = 9\)- Second row, second column: \((2)(1.6) + (3)(-1.4) = 3.2 - 4.2 = -1\)Thus, \[ A(BC) = \begin{bmatrix} 0 & 1.6 \ 9 & -1 \end{bmatrix} \].

Key Concepts

Matrix Dimensionality

When working with matrices, understanding their dimensionality is crucial. Each matrix is described using dimensions in the form of "rows x columns". For example, a 2x3 matrix has 2 rows and 3 columns. Here are the dimensions of the matrices from the exercise:

• Matrix A: 2x2
• Matrix B: 2x3
• Matrix C: 3x2

The dimensionality directly affects how matrices can interact with one another. Primarily, it determines if you can multiply two matrices. Always check dimensions to know what operations can be performed.

Algebraic Operations

Matrix multiplication is a type of algebraic operation that is more complex than simple arithmetic. Unlike the multiplication of numbers, matrix multiplication is not commutative, meaning that the order you multiply matrices in matters. For instance, multiplying matrix A with matrix B (AB) might yield different results from multiplying B with A (BA). Matrix multiplication involves taking the dot product of rows and columns. Here's how it works:

• For the element in the first row and first column of the product, you multiply corresponding elements of the first row of the first matrix and the first column of the second matrix, and then add those values together.
• This process is repeated for each element in the resulting matrix.

This type of operation requires careful attention to detail, particularly when dealing with larger matrices.

Matrix Product Compatibility

Matrix product compatibility revolves around whether the dimensions of the two matrices allow them to be multiplied. The rule to remember is: the number of columns in the first matrix must equal the number of rows in the second matrix. In our exercise:

• Matrix B (2x3) and Matrix C (3x2) are compatible because the number of columns (3) in B equals the number of rows (3) in C.
• The resulting product matrix BC is (2x2), which means it can now be multiplied by another 2x2 matrix, like Matrix A.

Product compatibility ensures that the operations you plan on executing can be performed without errors, making it a critical concept in matrix algebra.