Problem 25 - 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. $$ A=\left[\begin{array}{rr}{2} & {-5} \\ {6} & {7}\end{array}\right], B=\left[\begin{array}{cc}{-9} & {6} \\ {-4} & {2}\end{array}\right], C=\left[\begin{array}{cc}{0} & {9} \\ {7} & {1}\end{array}\right], D=\left[\begin{array}{rrr}{-8} & {7} & {-5} \\ {4} & {3} & {2} \\ {0} & {9} & {2}\end{array}\right], E=\left[\begin{array}{rrr}{4} & {5} & {3} \\ {7} & {-6} & {-5} \\ {1} & {0} & {9}\end{array}\right] $$ $$ 4 A+5 D $$

Step-by-Step Solution

Answer

The operation is not possible due to incompatible dimensions.

Step 1: Determine Matrix Compatibility

First, let's verify the sizes of the matrices involved in the operation. Matrix \( A \) is a \( 2 \times 2 \) matrix and matrix \( D \) is a \( 3 \times 3 \) matrix. Since they are not of the same dimensions, direct addition is not possible. However, since the operation involves scalar multiplication first, the incompatibility will not prevent us from proceeding with scalar multiplications.

Step 2: Perform Scalar Multiplication of Matrix A

Multiply each element of matrix \( A \) by 4:\[4A = 4 \times \left[\begin{array}{rr}{2} & {-5} \ {6} & {7}\end{array}\right] = \left[\begin{array}{rr}{8} & {-20} \ {24} & {28}\end{array}\right]\]

Step 3: Perform Scalar Multiplication of Matrix D

Multiply each element of matrix \( D \) by 5:\[5D = 5 \times \left[\begin{array}{rrr}{-8} & {7} & {-5} \ {4} & {3} & {2} \ {0} & {9} & {2}\end{array}\right] = \left[\begin{array}{rrr}{-40} & {35} & {-25} \ {20} & {15} & {10} \ {0} & {45} & {10}\end{array}\right]\]

Step 4: Combine the Results

In this exercise, the directive appears to be "add the scaled results," but direct addition is not possible as the matrices have different dimensions. Therefore, simply note that both results can be computed independently, but they cannot be added together as originally suggested:\( 4A = \left[\begin{array}{rr}{8} & {-20} \ {24} & {28}\end{array}\right] \)\( 5D = \left[\begin{array}{rrr}{-40} & {35} & {-25} \ {20} & {15} & {10} \ {0} & {45} & {10}\end{array}\right] \)

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental concept in matrix operations, where each element of a matrix is multiplied by a constant value, known as the scalar. This operation is straightforward and highly useful in various mathematical applications.
For instance, if we have a matrix \( A \) and a scalar value \( c \), then scalar multiplication involves multiplying each element of \( A \) by \( c \). Given a matrix \( A = \left[\begin{array}{cc} 2 & -5 \ 6 & 7 \end{array}\right] \), when we multiply it by the scalar 4, every element is transformed by this factor, producing \( 4A = \left[\begin{array}{cc} 8 & -20 \ 24 & 28 \end{array}\right] \).
This operation is particularly useful as it allows for the scaling or adjusting of matrices without altering their overall structure or pattern.

Matrix Addition

Matrix addition involves combining two matrices by adding their corresponding elements. However, this can only be done when the matrices involved share the same dimensions.
Consider matrices \( M \) and \( N \) of size \( m \times n \). Each element \( m_{ij} \) from matrix \( M \) and \( n_{ij} \) from matrix \( N \) are summed to obtain a new matrix \( P \) where \( p_{ij} = m_{ij} + n_{ij} \). This operation is crucial in many applications, including transforming data sets and adding effects in image processing.
However, one must always check the dimensions first, as two matrices can only be added if they are of the same size, otherwise the operation is undefined. In our exercise, matrix \( 4A \) has a different dimension from \( 5D \), thus making their direct addition impossible.

Matrix Dimension Compatibility

Matrix dimension compatibility is a critical factor that dictates whether certain matrix operations, like addition or subtraction, can be performed. It ensures that each element of one matrix has a corresponding element in the other matrix.
To exemplify, if you have one matrix with dimensions \( p \times q \), the second matrix must also be of dimensions \( p \times q \) for direct addition to be possible. If the dimensions do not match, the matrices are considered incompatible for addition or subtraction.
In the given exercise, we observed that matrix \( A \) is \( 2 \times 2 \) and matrix \( D \) is \( 3 \times 3 \). As their dimensions are different, it prevents any direct addition of these matrices, underscoring the importance of verifying compatibility before attempting operations.