Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Find \(A B\) $$ A=\left[\begin{array}{rr} 4 & -2 \\ 0 & 3 \\ -7 & 5 \end{array}\right], \quad B=\left[\begin{array}{l} 3 \\ 4 \end{array}\right] $$

Step-by-Step Solution

Answer

Matrix AB is \[ AB = \begin{bmatrix} 4 \\ 12 \\ -1 \end{bmatrix} \]

Step 1: Verify Matrix Dimensions

Check that the matrix multiplication can be performed by comparing the inner dimensions of the matrices. Matrix A is 3x2 and matrix B is 2x1, meaning the inner dimensions are compatible (2 and 2).

Step 2: Setup Matrix Multiplication

Multiply each element in the rows of matrix A by the corresponding element in the columns of matrix B to find the elements in the resulting matrix AB.

Step 3: Compute Each Element of AB

Compute the product of the first row of A with B: \(4*3 + (-2)*4 = 12 - 8 = 4\). Do the same for the second and third rows: \(0*3 + 3*4 = 12\) and \(-7*3 + 5*4 = -21 + 20 = -1\).

Step 4: Write the Resulting Matrix

Based on the computations, matrix AB is a 3x1 matrix: \[ AB = \left[ \begin{array}{c} 4 \ 12 \ -1 \end{array} \right] \]

Key Concepts

Understanding Matrix Dimensions

When multiplying matrices, the dimensions of the matrices play a crucial role in determining whether they can be multiplied together. Each matrix has dimensions described by the number of rows and columns it consists of. For instance, matrix \(A\) has dimensions 3x2, meaning it has 3 rows and 2 columns, while matrix \(B\) is 2x1, with 2 rows and 1 column.

The key to matrix multiplication is that the number of columns in the first matrix (matrix \(A\) in this case) must match the number of rows in the second matrix (matrix \(B\)). This is often referred to as the 'inner dimensions'. If these inner dimensions match, the matrices can be multiplied. The resulting matrix will have dimensions formed by the outer dimensions: in this case, 3 from matrix \(A\) and 1 from matrix \(B\), resulting in a 3x1 matrix. Understanding these dimensions is crucial to correctly performing matrix multiplication.

Matrix Multiplication Steps

The process of matrix multiplication involves a series of steps to ensure accuracy. It is not simply element-by-element multiplication, as in regular multiplication.

Here are the steps to multiply matrices:

• Set up the multiplication by aligning the matrices appropriately, ensuring the inner dimensions match.
• Multiplication involves taking each row from the first matrix and each column from the second matrix.
• For each element in the resulting matrix, multiply each element in the row by the corresponding element in the column and sum these products.

In the exercise example, multiply each element in a row of matrix \(A\) by the corresponding element in the column of matrix \(B\), then sum the results. This process is repeated for each row in matrix \(A\) to form the final matrix.

Computing the Resulting Matrix

Once the multiplication steps have been set up, computing each element involves simple arithmetic, but doing so correctly is key to obtaining the correct resulting matrix. Each element in the resulting matrix is a sum of products from rows and columns of the original matrices.

For example, the first element of the resulting matrix is calculated as follows:

• Take the first row of matrix \(A\): \([4, -2]\)
• Take the single column of matrix \(B\): \([3, 4]\)
• Compute the sum: \(4\times 3 + (-2)\times 4 = 12 - 8 = 4\)

Repeat this process for each row in matrix \(A\) using the column from matrix \(B\), resulting in the final matrix:

• For the second row \([0, 3]\): \(0\times 3 + 3\times 4 = 0 + 12 = 12\)
• For the third row \([-7, 5]\): \(-7\times 3 + 5\times 4 = -21 + 20 = -1\)

The resulting matrix \(AB\) is then a 3x1 matrix: \[ \begin{bmatrix} 4 \ 12 \ -1 \end{bmatrix} \] Making sure to follow these steps will ensure accurate matrix multiplication.