Problem 23 - Algebra for College Students

Question

For Problems \(19-26\), compute \(A B\). $$ A=\left[\begin{array}{rr} -4 & 2 \\ 7 & -5 \end{array}\right], \quad \bar{B}=\left[\begin{array}{l} -1 \\ -4 \end{array}\right] $$

Step-by-Step Solution

Answer

The product matrix is \( \begin{bmatrix} -4 \\ 13 \end{bmatrix} \).

Step 1: Verify Compatibility for Multiplication

Matrix multiplication can only be performed when the number of columns in the first matrix matches the number of rows in the second matrix. Here, matrix \( A \) is a 2x2 matrix and matrix \( B \) is a 2x1 matrix. Since both have 2 as the dimension to match (columns of \( A \) and rows of \( B \)), they can be multiplied.

Step 2: Set Up the Multiplication

The product \( AB \) will result in a new matrix with dimensions derived from the number of rows in \( A \) and columns in \( B \). Therefore, the resulting matrix will be a 2x1 matrix. Denote this resulting matrix as \( C = \begin{bmatrix} c_1 \ c_2 \end{bmatrix} \).

Step 3: Compute the First Element of Resulting Matrix

Calculate the first element \( c_1 \) of the matrix \( C \) by performing: \((-4) \times (-1) + 2 \times (-4)\). This simplifies to \(4 - 8 = -4\). Therefore, \( c_1 = -4 \).

Step 4: Compute the Second Element of Resulting Matrix

Calculate the second element \( c_2 \) of the matrix \( C \) by performing: \(7 \times (-1) + (-5) \times (-4)\). This simplifies to \(-7 + 20 = 13\). Therefore, \( c_2 = 13 \).

Step 5: Write the Final Result Matrix

The resulting matrix \( C \) after multiplying \( A \) and \( B \) is \( \begin{bmatrix} -4 \ 13 \end{bmatrix} \). The operation is complete.

Key Concepts

Matrix Compatibility

Before you can multiply two matrices, it's vital to ensure they are compatible. Matrix compatibility refers to the condition necessary for performing matrix multiplication. This is determined by the dimensions of the matrices involved. You need to check the number of columns in the first matrix and compare it to the number of rows in the second matrix.

• If they match, the matrices are compatible, and multiplication is possible.
• If they do not match, you cannot proceed with multiplication.

For example, if matrix A has 2 columns and matrix B has 2 rows, then these matrices are compatible for multiplication.

Matrix Dimensions

Understanding matrix dimensions is crucial for performing matrix operations. The dimensions of a matrix are written as 'rows x columns'. For instance, a 2x3 matrix has 2 rows and 3 columns. This notation provides a detailed structure of the matrix format.
In the context of multiplication:

• The first matrix's columns must equal the second matrix's rows.
• The dimensions of the resultant matrix are determined by the number of rows in the first matrix and the number of columns in the second matrix.

With matrices A (2x2) and B (2x1) as our example, we can see that our resultant matrix will have dimensions 2x1.

Matrix Computation

Matrix computation involves calculating each element of the resulting matrix after verifying compatibility. You do this by carrying out specific arithmetic operations on the elements.
Each element in the resultant matrix is computed as the sum of products from the corresponding row elements of the first matrix and column elements of the second matrix.
For instance,

• The first element of the resultant matrix is calculated using the first row of matrix A and the first column of matrix B.
• Similarly, the second element uses the second row of A and the same column of B, given B is a single-column matrix.

Always remember, matrix multiplication is not like regular multiplication, and the order of multiplication matters.

Matrix Resultant Matrix

Once the computation completes, you'll get the resultant matrix. This matrix represents the outcome of the matrix multiplication. The dimensions of this resultant matrix have already been established based on the initial dimensions of the matrices involved.
Using our matrices A (2x2) and B (2x1):

• The resultant matrix will be 2x1.
• Each element in this matrix corresponds to a unique combination of calculations from A and B.
• In our example, the resultant matrix, \( C \), ends up being \( \begin{bmatrix} -4 \ 13 \end{bmatrix} \), representing new values derived from the combined data of A and B.

This resultant matrix holds significant value in applications where matrix multiplication is integral, such as transformations in geometry or solving systems of linear equations.