Question

Use the following matrices. Determine whether the given expression is defined. If it is defined, express the result as a single matrix; if it is not, write "not defined" $$ A=\left[\begin{array}{rrr} 0 & 3 & -5 \\ 1 & 2 & 6 \end{array}\right] \quad B=\left[\begin{array}{rrr} 4 & 1 & 0 \\ -2 & 3 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 4 & 1 \\ 6 & 2 \\ -2 & 3 \end{array}\right] $$ BC

Short answer

Matrix BC = \( \left[ \begin{array}{rr} 22 & 6 \ 14 & -2 \end{array} \right] \)

Step-by-step solution

- Determine the dimensions of matrices B and C

Matrix B is a 2x3 matrix with dimensions 2 rows and 3 columns. Matrix C is a 3x2 matrix with dimensions 3 rows and 2 columns.

- Check the compatibility for multiplication

For two matrices B (2x3) and C (3x2) to be multiplied, the number of columns in the first matrix (B) must be equal to the number of rows in the second matrix (C). Here, matrix B has 3 columns and matrix C has 3 rows, so they are compatible for multiplication.

- Perform the matrix multiplication BC

To find the product of BC, compute each element of the resulting matrix by taking the dot product of the corresponding row from matrix B and the corresponding column from matrix C.Let's calculate each element step-by-step:\[BC = \left[\begin{array}{rrr}4 & 1 & 0 \-2 & 3 & -2\end{array}\right] \left[\begin{array}{rr}4 & 1 \6 & 2 \-2 & 3\end{array}\right]\]Element (1,1): \((B_{11} \times C_{11} + B_{12} \times C_{21} + B_{13} \times C_{31})\) = \((4 \times 4 + 1 \times 6 + 0 \times -2)\) = \(16 + 6 + 0 = 22\)Element (1,2): \((B_{11} \times C_{12} + B_{12} \times C_{22} + B_{13} \times C_{32})\) = \((4 \times 1 + 1 \times 2 + 0 \times 3)\) = \(4 + 2 + 0 = 6\)Element (2,1): \((B_{21} \times C_{11} + B_{22} \times C_{21} + B_{23} \times C_{31})\) = \((-2 \times 4 + 3 \times 6 + -2 \times -2)\) = \(-8 + 18 + 4 = 14\)Element (2,2): \((B_{21} \times C_{12} + B_{22} \times C_{22} + B_{23} \times C_{32})\) = \((-2 \times 1 + 3 \times 2 + -2 \times 3)\) = \(-2 + 6 - 6 = -2\)

- Write the resulting matrix

The resulting matrix after multiplying B and C is:\[BC = \left[\begin{array}{rr}22 & 6 \14 & -2\end{array}\right]\]

Key concepts

Matrix Dimensions

Understanding the dimensions of matrices is crucial for performing operations like matrix multiplication. Dimensions of a matrix are denoted as 'rows x columns'. For example, a matrix with 2 rows and 3 columns is termed as a 2x3 matrix. In the example provided, matrix B is a 2x3 matrix, and matrix C is a 3x2 matrix. Recognizing these dimensions helps you understand how the matrices can interact with each other.

Matrix Compatibility

Matrix compatibility is essential for determining whether two matrices can be multiplied. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. In simpler terms, if matrix B is 2x3 and matrix C is 3x2, these matrices are compatible for multiplication because the number of columns in B (which is 3) matches the number of rows in C (which is also 3). This ensures that each element in a row of B can appropriately interact with each element in a column of C.

Dot Product

The dot product is a foundational arithmetic operation when computing matrix multiplication. It's the sum of the products of corresponding elements from two sequences of numbers. In matrix multiplication, we find the dot product of rows of the first matrix and columns of the second matrix.

For example, to compute the element in the first row and first column of the resulting matrix from multiplying matrix B and matrix C:

• Take the first row of B: [4, 1, 0]
• Take the first column of C: [4, 6, -2]
• Compute: (4 * 4) + (1 * 6) + (0 * -2) = 16 + 6 + 0 = 22

This process is repeated for each element in the resulting matrix.

Matrix Multiplication Steps

Matrix multiplication involves systematic steps that ensure accurate results:

• Identify the dimensions of the matrices involved, confirming their compatibility.
• Compute the dot product for each element in the resulting matrix by taking the corresponding row from the first matrix and the corresponding column from the second matrix.
• Write down each calculated value in its respective position in the resulting matrix.

In our example, we performed dot products for four elements resulting in the values 22, 6, 14, and -2. Detailed calculations for each element were provided, showing the step-by-step process to aid understanding.

Resulting Matrix

The final step in matrix multiplication is compiling the computed values into the resulting matrix. After performing the necessary calculations for our example, we derived the following values:

• Element (1,1): 22
• Element (1,2): 6
• Element (2,1): 14
• Element (2,2): -2

Consequently, the resulting matrix from multiplying matrices B and C is:
\[\begin{equation} \begin{bmatrix} 22 & 6 \ 14 & -2 \ \ \end{bmatrix} \ \ \end{equation}\]