Problem 50 - 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. Use a calculator to verify your solution. \(A=\left[\begin{array}{rrr}-2 & 0 & 9 \\ 1 & 8 & -3 \\ 0.5 & 4 & 5\end{array}\right], B=\left[\begin{array}{rrr}0.5 & 3 & 0 \\ -4 & 1 & 6 \\\ 8 & 7 & 2\end{array}\right], C=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\\ 1 & 0 & 1\end{array}\right]\) \(A B\)

Step-by-Step Solution

Answer

Matrix multiplication is possible, and the product is \(\begin{bmatrix} 71 & 57 & 18 \\ -55.5 & -10 & 42 \\ 24.25 & 40.5 & 34 \end{bmatrix}\).

Step 1: Determine the dimensions

First, examine the dimensions of the matrices \(A\) and \(B\). Matrix \(A\) is a \(3 \times 3\) matrix and matrix \(B\) is also a \(3 \times 3\) matrix. Since the number of columns in \(A\) is equal to the number of rows in \(B\), the multiplication \(AB\) is possible.

Step 2: Matrix multiplication setup

To multiply two matrices \(A = [a_{ij}]\) and \(B = [b_{ij}]\), compute the element in the \(i\)-th row and \(j\)-th column of the product matrix \(AB\) as follows: \((AB)_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + a_{i3}b_{3j}\).

Step 3: Calculate the first row of AB

Compute each element in the first row of \(AB\):- \((AB)_{11} = (-2)(0.5) + (0)(-4) + (9)(8) = -1 + 0 + 72 = 71\)- \((AB)_{12} = (-2)(3) + (0)(1) + (9)(7) = -6 + 0 + 63 = 57\)- \((AB)_{13} = (-2)(0) + (0)(6) + (9)(2) = 0 + 0 + 18 = 18\)

Step 4: Calculate the second row of AB

Compute each element in the second row of \(AB\):- \((AB)_{21} = (1)(0.5) + (8)(-4) + (-3)(8) = 0.5 - 32 - 24 = -55.5\)- \((AB)_{22} = (1)(3) + (8)(1) + (-3)(7) = 3 + 8 - 21 = -10\)- \((AB)_{23} = (1)(0) + (8)(6) + (-3)(2) = 0 + 48 - 6 = 42\)

Step 5: Calculate the third row of AB

Compute each element in the third row of \(AB\):- \((AB)_{31} = (0.5)(0.5) + (4)(-4) + (5)(8) = 0.25 - 16 + 40 = 24.25\)- \((AB)_{32} = (0.5)(3) + (4)(1) + (5)(7) = 1.5 + 4 + 35 = 40.5\)- \((AB)_{33} = (0.5)(0) + (4)(6) + (5)(2) = 0 + 24 + 10 = 34\)

Step 6: Compile the result

After calculating each element, combine them to form the product matrix \(AB\):\[AB = \begin{bmatrix} 71 & 57 & 18 \ -55.5 & -10 & 42 \ 24.25 & 40.5 & 34 \end{bmatrix}\]

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is essential for performing operations like matrix multiplication. The dimensions of a matrix are given in terms of rows and columns, written as "rows × columns." For example, matrix \( A \) is a \( 3 \times 3 \) matrix, meaning it has 3 rows and 3 columns. Similarly, matrix \( B \) is also \( 3 \times 3 \).

The rule for multiplying two matrices is that the number of columns in the first matrix must match the number of rows in the second matrix. Therefore, if matrix \( A \) has dimensions \( m \times n \) and matrix \( B \) has \( n \times p \), then the product matrix \( AB \) will have dimensions \( m \times p \).

In our example, both matrices \( A \) and \( B \) are square matrices of the same dimensions, \( 3 \times 3 \), so their multiplication is straightforwardly possible.

Element Computation

Element computation in matrix multiplication requires finding each element of the resulting matrix by performing a series of multiplications and additions. For each element in the resulting matrix \( AB \), you focus on a specific row of matrix \( A \) and a specific column of matrix \( B \).

The formula for calculating the element \((AB)_{ij}\) in the resulting matrix is: \[(AB)_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + ... + a_{in}b_{nj}\] Where \( a_{ik} \) are elements from the \( i\)-th row of matrix \( A \) and \( b_{kj} \) are elements from the \( j\)-th column of matrix \( B \).

For instance, to compute \( (AB)_{11} \), multiply each element of the first row of \( A \) with the corresponding element of the first column of \( B \), and then sum up these products. This process is repeated for each element of the product matrix. Breaking down computations into these steps makes it easier to verify results and ensures accuracy.

Matrix Operations

Matrix operations encompass more than just matrix multiplication. They include addition, subtraction, and scalar multiplication as well. However, in this case, we are discussing multiplication.

Besides the dimensional compatibility requirement, matrices need to be "conformable" for addition and subtraction, meaning they must have the same dimensions. For multiplication, as outlined, only the number of columns in the first matrix must match the number of rows in the second.

Interestingly, matrix multiplication is not commutative; in simple terms, \( AB \) doesn't necessarily equal \( BA \). This is a critical property to remember, as reversing the multiplication order often changes the resultant matrix. Knowing these properties helps in setting the right expectations when performing matrix operations.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. Matrix multiplication is a fundamental concept in linear algebra.

It allows transformations of space, helping solve systems of linear equations, among many other applications. When you multiply two matrices, you are essentially applying two linear transformations successively.

Understanding the properties of matrices and operations like multiplication is essential for exploring advanced topics in linear algebra, such as eigenvectors and eigenvalues, determinants, and matrix decompositions. Often, real-world problems in engineering, physics, computer science, and more can be modeled using these concepts, making them indispensable tools in mathematical and scientific analysis.