Problem 42 - College Algebra

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.(Hint: \(A^{2}=A \cdot A )\) $$ A=\left[\begin{array}{ll}{1} & {0} \\ {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{-2} & {3} & {4} \\ {-1} & {1} & {-5}\end{array}\right], C=\left[\begin{array}{rr}{0.5} & {0.1} \\ {1} & {0.2} \\\ {-0.5} & {0.3}\end{array}\right], D=\left[\begin{array}{rrr}{1} & {0} & {-1} \\ {-6} & {7} & {5} \\ {4} & {2} & {1}\end{array}\right] $$ $$ B A $$

Step-by-Step Solution

Answer

The operation \( BA \) is not possible because the dimensions are not compatible for multiplication.

Step 1: Confirm Matrix Dimensions

Matrix \( B \) is a \( 2 \times 3 \) matrix and matrix \( A \) is a \( 2 \times 2 \) matrix. For matrix multiplication to be defined, the number of columns in the first matrix \( B \) must match the number of rows in the second matrix \( A \). However, \( B \) has 3 columns and \( A \) has 2 rows, meaning their multiplication is not possible.

Step 2: Explain Matrix Multiplication Rules

Matrix multiplication is only possible when the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, \( B \) cannot be multiplied by \( A \) because the column-row dimension rule (3 columns in \( B \) vs. 2 rows in \( A \)) is not satisfied.

Step 3: Conclusion

Since the dimension requirements for multiplication are not satisfied, the operation \( BA \) is not possible. Therefore, \( BA \) cannot be computed.

Key Concepts

Matrix Dimensions

Understanding the dimensions of a matrix is crucial for performing matrix operations. Each matrix has its own dimensions, defined by the number of rows and columns it contains.
If you have a matrix with 2 rows and 3 columns, it is called a \(2 \times 3\) matrix. Similarly, if a matrix has 2 rows and 2 columns, it is called a \(2 \times 2\) matrix.

• The first number in the dimension notation represents the number of rows.
• The second number represents the number of columns.

Knowing matrix dimensions helps you verify whether certain matrix operations are possible or not, as dimensions are a key factor in these operations.

Matrix Operations

Matrix operations include addition, subtraction, scalar multiplication, and matrix multiplication. Each operation has specific requirements and rules.
Let's take a closer look:

• Addition and Subtraction: Two matrices can be added or subtracted only if they have the same dimensions. This means a \(3 \times 2\) matrix can only be added to or subtracted from another \(3 \times 2\) matrix.
• Scalar Multiplication: You can multiply a matrix by a constant (scalar), which simply means multiplying each element of the matrix by that constant.
• Matrix Multiplication: Involves particular rules as we'll see in the next section. The key requirement for multiplication is matching dimensions.

Understanding these operations and their requirements helps avoid mistakes when working with matrices.

Matrix Multiplication Rules

Matrix multiplication is a bit more complex compared to other operations. Let's break it down:

• Dimension Matching: For two matrices to be multiplied, the number of columns in the first matrix needs to equal the number of rows in the second. For example, a \(2 \times 3\) matrix can be multiplied by a \(3 \times 2\) matrix.
• Result Dimensions: The product of the matrices will have dimensions defined by the rows of the first matrix and the columns of the second. In the previous example, the result would be a \(2 \times 2\) matrix.
• Element Calculation: The elements in the resulting matrix are calculated by taking the dot product of corresponding row vectors from the first matrix and column vectors from the second.

In our exercise example, matrix \(B\) with dimensions \(2 \times 3\) cannot be multiplied by matrix \(A\), which has dimensions \(2 \times 2\), because the number of columns in \(B\) does not match the number of rows in \(A\). Understanding these rules prevents mistakes in matrix operations.