Problem 21 - College Algebra

Question

For the following exercises, use the matrices below to perform matrix multiplication. $$ A=\left[\begin{array}{rr}{-1} & {5} \\ {3} & {2}\end{array}\right], B=\left[\begin{array}{rrr}{3} & {6} & {4} \\ {-8} & {0} & {12}\end{array}\right], C=\left[\begin{array}{rr}{4} & {10} \\ {-2} & {6} \\\ {5} & {9}\end{array}\right], D=\left[\begin{array}{rrr}{2} & {-3} & {12} \\\ {9} & {3} & {1} \\ {0} & {8} & {-10}\end{array}\right] $$ $$ B D $$

Step-by-Step Solution

Answer

The product matrix BD is \(\begin{bmatrix}60 & 41 & 2 \\ -16 & 120 & -216\end{bmatrix}\)

Step 1: Understand Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix. Ensure that the number of columns in the first matrix matches the number of rows in the second matrix. Since matrix \(B\) is \(2 \times 3\) and matrix \(D\) is \(3 \times 3\), they can be multiplied to yield a \(2 \times 3\) matrix.

Step 2: Multiply Matrices B and D

Calculate each element of the result matrix by multiplying the corresponding row of matrix \(B\) with the column of matrix \(D\). For result matrix element at position (1,1): \((3\times2) + (6\times9) + (4\times0)\). For element (1,2): \((3\times-3) + (6\times3) + (4\times8)\). For element (1,3): \((3\times12) + (6\times1) + (4\times-10)\). Repeat this process for the second row of matrix \(B\).

Step 3: Calculate First Row of Result

Calculate each element:- (1,1): \(6 + 54 + 0 = 60\).- (1,2): \(-9 + 18 + 32 = 41\).- (1,3): \(36 + 6 - 40 = 2\).The first row of the product matrix is \([60, 41, 2]\).

Step 4: Calculate Second Row of Result

Calculate each element:- (2,1): \(-8\times2 + 0\times9 + 12\times0 = -16\).- (2,2): \(-8\times-3 + 0\times3 + 12\times8 = 120\).- (2,3): \(-8\times12 + 0\times1 + 12\times-10 = -216\).The second row of the product matrix is \([-16, 120, -216]\).

Step 5: Write the Final Result Matrix

The resulting matrix from multiplying \(B\) by \(D\) is:\[\begin{bmatrix}60 & 41 & 2 \-16 & 120 & -216\end{bmatrix}\]

Key Concepts

Matrix Operations

Matrix operations are fundamental procedures in linear algebra, allowing us to manipulate arrays of numbers systematically. One of the essential operations in matrices is multiplication. This operation follows specific rules:

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• Each entry in the resulting matrix is determined by multiplying corresponding entries from the first matrix's row and second matrix's column, and then summing these products.

Matrix multiplication is not commutative, meaning that the order in which matrices are multiplied affects the result. Therefore, multiplying matrix A by matrix B yields a different product than multiplying matrix B by matrix A. Understanding these rules is crucial for correctly performing matrix operations, such as those necessary for solving systems of equations and transforming geometrical entities.

Matrix Dimensions

Matrix dimensions convey the size of a matrix, providing information about how many rows and columns it has. This is denoted as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns. For instance, a \(2 \times 3\) matrix has 2 rows and 3 columns.

In matrix multiplication, dimensions play a critical role. To multiply two matrices, the number of columns in the first must equal the number of rows in the second, making their alignment crucial. The result of multiplying a \(2 \times 3\) matrix with a \(3 \times 3\) matrix would be a \(2 \times 3\) matrix, as the inner dimensions (columns of the first and rows of the second) must match, and the outer dimensions become those of the resulting matrix. Having clear knowledge of matrix dimensions helps in verifying whether multiplication of matrices is feasible.

Element-wise Multiplication

Element-wise multiplication, also known as the Hadamard product, differs significantly from standard matrix multiplication. It involves multiplying corresponding elements from matrices of the same size, element by element. For this operation, both matrices must have identical dimensions.

• The process: multiply the elements in the same position of each matrix to produce a new matrix.
• Every element from the first matrix is paired with the corresponding element in the second matrix for multiplication.

Element-wise multiplication applies directly to each pair of corresponding elements, unlike traditional matrix multiplication, which is more complex due to its combination of rows and columns. While matrix multiplication has strict dimension requirements, element-wise multiplication requires straightforward one-to-one correspondence between elements, making it a less error-prone operation for same-sized matrices.