Problem 48 - College Algebra

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

Step-by-Step Solution

Answer

The result of \((AB)C\) is \(\begin{bmatrix} 0 & 1.6 \\ 9 & -1 \end{bmatrix}\).

Step 1: Verify Compatiblity of (AB)

First, we need to determine if matrices \(A\) and \(B\) are compatible for multiplication. Matrix \(A\) is a \(2 \times 2\) matrix and matrix \(B\) is a \(2 \times 3\) matrix. Matrix multiplication is possible when the number of columns in the first matrix (here, \(A\) has 2 columns) matches the number of rows in the second matrix (here, \(B\) has 2 rows). Thus, matrices \(A\) and \(B\) are compatible, and the product \(A \cdot B\) will result in a \(2 \times 3\) matrix.

Step 2: Calculate (AB)

Compute the product \(AB\) by taking the dot product of rows from \(A\) with columns from \(B\):\[AB = \begin{bmatrix} 1 & 0 \ 2 & 3 \end{bmatrix} \cdot \begin{bmatrix} -2 & 3 & 4 \ -1 & 1 & -5 \end{bmatrix}\]- Calculate the first row of \(AB\): - First row, first column: \((1 \times -2) + (0 \times -1) = -2\) - First row, second column: \((1 \times 3) + (0 \times 1) = 3\) - First row, third column: \((1 \times 4) + (0 \times -5) = 4\)- Calculate the second row of \(AB\): - Second row, first column: \((2 \times -2) + (3 \times -1) = -4 - 3 = -7\) - Second row, second column: \((2 \times 3) + (3 \times 1) = 6 + 3 = 9\) - Second row, third column: \((2 \times 4) + (3 \times -5) = 8 - 15 = -7\)Thus, \(AB = \begin{bmatrix} -2 & 3 & 4 \ -7 & 9 & -7 \end{bmatrix}\).

Step 3: Verify Compatibility of (AB)C

Next, check if the resulting matrix \(AB\) can be multiplied by matrix \(C\). Matrix \(AB\) is \(2 \times 3\) and matrix \(C\) is \(3 \times 2\). The multiplication is possible if the number of columns in \(AB\) (which is 3) is equal to the number of rows in \(C\) (which is 3). They match, so multiplication is possible, and the result will be a \(2 \times 2\) matrix.

Step 4: Calculate ((AB)C)

Compute the product \((AB)C\):\[(AB)C = \begin{bmatrix} -2 & 3 & 4 \ -7 & 9 & -7 \end{bmatrix} \cdot \begin{bmatrix} 0.5 & 0.1 \ 1 & 0.2 \ -0.5 & 0.3 \end{bmatrix}\]- Calculate the first row of \((AB)C\): - First row, first column: \((-2 \times 0.5) + (3 \times 1) + (4 \times -0.5) = -1 + 3 - 2 = 0\) - First row, second column: \((-2 \times 0.1) + (3 \times 0.2) + (4 \times 0.3) = -0.2 + 0.6 + 1.2 = 1.6\)- Calculate the second row of \((AB)C\): - Second row, first column: \((-7 \times 0.5) + (9 \times 1) + (-7 \times -0.5) = -3.5 + 9 + 3.5 = 9\) - Second row, second column: \((-7 \times 0.1) + (9 \times 0.2) + (-7 \times 0.3) = -0.7 + 1.8 - 2.1 = -1\)Thus, \((AB)C = \begin{bmatrix} 0 & 1.6 \ 9 & -1 \end{bmatrix}\).

Key Concepts

Matrix Compatibility

When performing matrix multiplication, ensuring matrix compatibility is crucial. This means that for two matrices to be compatible for multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. This rule is necessary because matrix multiplication involves performing operations between rows of the first matrix and columns of the second matrix.

For example, consider matrices A and B from the problem. Matrix A is a 2x2 matrix, meaning it has 2 rows and 2 columns, and matrix B is a 2x3 matrix, having 2 rows and 3 columns. Here, the number of columns in A (2) matches the number of rows in B (2), making them compatible for multiplication. Thus, we can calculate the product A \( \cdot \) B, resulting in a new matrix with dimensions derived from the outer dimensions of the original matrices.

Dot Product

The dot product is fundamental in matrix multiplication. It involves multiplying the corresponding entries of a row in the first matrix by the entries of a column in the second matrix, then summing the products. This calculation provides each entry in the resulting matrix.

As an example, when calculating the dot product for the entry at the first row, first column of the product matrix AB, we took the entries of the first row of A \([1, 0]\) and the first column of B \([-2, -1]\). We computed the dot product as \((1 \times -2) + (0 \times -1) = -2\). This sum then became the value in the first row, first column of the matrix AB. This process is repeated for each corresponding row and column pairing to complete the matrix.

Matrix Dimensions

Understanding matrix dimensions is key to successfully using matrices. Each matrix is defined by its dimensions, denoted as rows x columns. These dimensions guide both the process and outcome of matrix operations.

In the original exercise, matrix A is 2x2, while matrix B is 2x3. After multiplication, the resulting matrix AB is 2x3 because the number of rows comes from A and the number of columns comes from B.

• Matrix A: 2x2 - 2 rows, 2 columns
• Matrix B: 2x3 - 2 rows, 3 columns
• Result of A \( \cdot \) B: 2x3 - 2 rows, 3 columns

Furthermore, understanding how dimensions change with operations helps predict the size and structure of the resulting matrix, which is vital for planning further operations.

Matrix Operations

Matrix operations include a variety of procedures such as addition, subtraction, and multiplication. Multiplication is unique because, unlike addition and subtraction (where matrices must have the same dimensions), it requires compatibility in a different sense.

The operation of multiplying matrices A and B was carried out through dot products due to their compatible dimensions. This process resulted in a new matrix that could be utilized for further operations, like multiplying the product AB by another matrix C. Operations might differ in rules but share the fundamental aim of transforming data held within matrices into usable outputs for various mathematical and practical applications.