Problem 43 - Precalculus

Question

perform the indicated operations for each expression, if possible. $$A=\left[\begin{array}{rrr}-1 & 3 & 0 \\\2 & 4 & 1\end{array}\right] \quad B=\left[\begin{array}{rr}0 & 2 & 1 \\\3 & -2 & 4\end{array}\right] \quad C=\left[\begin{array}{rr}0 & 1 \\\2 & -1 \\\3 & 1 \end{array}\right] \quad D=\left[\begin{array}{rr}2 & -3 \\\0 & 1 \\\4 & -2 \end{array}\right]$$ $$E=\left[\begin{array}{rrr}-1 & 0 & 1 \\\2 & 1 & 4 \\\\-3 & 1 & 5 \end{array}\right] \quad F=\left[\begin{array}{r}1 \\\0 \\\\-1\end{array}\right] \quad G=\left[\begin{array}{ll}1 & 2 \\\3 & 4\end{array}\right]$$ $$D F$$

Step-by-Step Solution

Answer

Matrix multiplication \(DF\) is not possible due to incompatible dimensions.

Step 1: Define the Operation

We need to multiply matrix \(D\) by matrix \(F\). Matrix \(D\) is a \(3\times 2\) matrix and matrix \(F\) is a \(3\times 1\) vector. For the multiplication to be possible, the number of columns in the first matrix must equal the number of rows in the second matrix.

Step 2: Check Dimensional Compatibility

Matrix \(D\) has dimensions \(3\times 2\), and matrix \(F\) has dimensions \(3\times 1\). Since the number of columns in \(D\) (which is 2) does not equal the number of rows in \(F\) (which is 3), the matrices \(D\) and \(F\) cannot be multiplied together.

Key Concepts

Matrix Dimensions

When we talk about matrices, one of the first things to consider is their dimensions. The dimensions of a matrix are given by the number of rows multiplied by the number of columns, written as "rows \(\times\) columns". Knowing the dimensions is crucial because it determines whether certain matrix operations, such as multiplication, can be carried out.

For example, let's examine matrices \( D \) and \( F \) from the exercise. Matrix \( D \) is a \(3 \times 2\) matrix, which means it has 3 rows and 2 columns. Meanwhile, matrix \( F \) is a \(3 \times 1\) matrix, meaning it has 3 rows and 1 column.

• Matrix \( D \): 3 rows, 2 columns
• Matrix \( F \): 3 rows, 1 column

Understanding the dimensions is the first step in matrix arithmetic, guiding what operations are possible.

Matrix Operations

Matrix operations include addition, subtraction, multiplication, and more. To perform matrix operations, especially multiplication, understanding how these operations work is essential. Let's focus on matrix multiplication because it's a bit more complex than addition or subtraction.

In matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix. This is necessary because the operation involves computing dot products between rows of the first matrix and columns of the second matrix. This process doesn't work if the dimensions aren't compatible. When matrices have compatible dimensions, the resulting product is a new matrix. Its dimensions are defined by the number of rows from the first matrix and the number of columns from the second matrix.

For our exercise, attempting to multiply \( D \) (\(3 \times 2\)) and \( F \) (\(3 \times 1\)) would require matching the 2 columns of \( D \) with 3 rows of \( F \), which isn't possible. Therefore, no product matrix can be formed in this scenario.

Matrix Compatibility

Matrix compatibility is a fundamental concept in matrix operations, especially multiplication. Compatibility determines whether two matrices can be multiplied together. To multiply a matrix \( A \) and a matrix \( B \), \( A \) must have as many columns as \( B \) has rows.

Let's relate this back to our example from the original problem. We attempted to multiply matrices \( D \) and \( F \):

• Matrix \( D \) dimensions: \(3 \times 2\)
• Matrix \( F \) dimensions: \(3 \times 1\)

Here, matrix \( F \) has 3 rows, whereas matrix \( D \) has only 2 columns. Since these numbers don't match, the matrices aren't compatible for multiplication. This is why understanding compatibility is crucial before attempting matrix multiplication—it saves time and prevents errors.

Remember that if dimensions are compatible, you proceed with matrix multiplication, resulting in a new matrix. The dimension of this new matrix will be defined by the number of rows from the first matrix and the number of columns from the second matrix.