Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Find \(A B\), if possible. $$ A=\left[\begin{array}{rr} 3 & -2 \\ 4 & 5 \\ 1 & -1 \end{array}\right], B=\left[\begin{array}{rrrr} -1 & 4 & -2 & 5 \\ 2 & 1 & 3 & -1 \end{array}\right] $$

Step-by-Step Solution

Answer

So, the product of the matrix \(A\) and \(B\)— i.e., \(AB\)— is: \[\left[\begin{array}{rrrr} -7 & 10 & 0 & 11 \ 6 & 21 & 7 & 17 \ -3 & 3 & -1 & 6 \end{array}\right]\]

Step 1: Checking Compatibility

The matrix \(A\) has shape \(3x2\) and the matrix \(B\) has shape \(2x4\). For matrix multiplication, the number of columns in the first matrix should be equal to the number of rows in the second matrix. In this case, the column of the first matrix \(A\) (which is 2) is equal to the row of the second matrix \(B\) (which is also 2). Hence we can perform multiplication.

Step 2: Implementing Matrix Multiplication

Matrix multiplication involves taking the dot product of the rows of the first matrix with the columns of the second. So, let's multiply each row in \(A\) with each column in \(B\) and take sums. The result of the multiplication will be a matrix of shape \(3x4\) (number of rows in \(A\) by number of columns in \(B\)). For example, element at position (1,1) in resulting matrix will be a dot product of first row of matrix \(A\) and first column of matrix \(B\) i.e. \((3*-1) + (-2*2)=-7\). Do this for all rows in \(A\) and columns in \(B\).

Step 3: Calculating the Result

Using the steps described above, the result will be: \[ AB=\left[\begin{array}{rrrr} -7 & 10 & 0 & 11 \ 6 & 21 & 7 & 17 \ -3 & 3 & -1 & 6 \end{array}\right]\]

Key Concepts

Matrix Compatibility

Before diving into matrix multiplication, it's crucial to discuss matrix compatibility. Matrix compatibility refers to the ability to multiply two matrices. This depends on the dimensions of these matrices.

• A matrix is defined by the number of rows and columns it contains.
• 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.

Consider matrices \(A\) and \(B\) from the original exercise:

• Matrix \(A\) has dimensions \(3 \times 2\) (3 rows and 2 columns).
• Matrix \(B\) has dimensions \(2 \times 4\) (2 rows and 4 columns).

In this case, the matrices are compatible because the 2 columns in \(A\) match the 2 rows in \(B\). Hence, multiplying \(A\) and \(B\) to get AB is possible.

Dot Product

The dot product is a key concept in matrix multiplication. It involves calculating the sum of the products of corresponding elements from one row and one column.Here's how to understand the dot product in the context of our matrices:

• Take a row from the first matrix.
• Take a column from the second matrix.
• Multiply the corresponding elements from the row and column and then sum these products.

For example, the first element in the resulting matrix from \(A\) and \(B\) multiplication (position (1,1)) is found by:- Row 1 of \(A\): \([3, -2]\)- Column 1 of \(B\): \([-1, 2]\)- Calculation: \(3 \times (-1) + (-2) \times 2 = -7\)This dot product operation is repeated for all combinations of rows from matrix \(A\) and columns from matrix \(B\) to fill the resulting matrix.

Matrix Dimensions

Understanding matrix dimensions helps you predict the size of the resulting matrix after multiplication.In our example:

• The first matrix \(A\) is \(3 \times 2\), meaning it has 3 rows and 2 columns.
• The second matrix \(B\) is \(2 \times 4\), with 2 rows and 4 columns.

When you multiply these matrices, the resulting matrix will have the dimensions of \(3 \times 4\). This is because:- The number of rows in the resulting matrix comes from the first matrix \(A\).- The number of columns in the resulting matrix comes from the second matrix \(B\).Knowing this, you can easily determine that the resulting matrix has 3 rows and 4 columns, making it possible to plan your calculations accordingly.