Question

Use the following matrices. Determine whether the given expression is defined. If it is defined, express the result as a single matrix; if it is not, write "not defined" $$ A=\left[\begin{array}{rrr} 0 & 3 & -5 \\ 1 & 2 & 6 \end{array}\right] \quad B=\left[\begin{array}{rrr} 4 & 1 & 0 \\ -2 & 3 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 4 & 1 \\ 6 & 2 \\ -2 & 3 \end{array}\right] $$ C(A+B)

Short answer

The expression C(A+B) is not defined.

Step-by-step solution

Check dimensions

First, establish the dimensions of each matrix. Matrix A is a 2x3 matrix (2 rows and 3 columns), matrix B is a 2x3 matrix, and matrix C is a 3x2 matrix.

Add matrices A and B

Since matrices A and B have the same dimensions, they can be added element-wise. \[ A + B = \begin{bmatrix} 0 & 3 & -5 \ 1 & 2 & 6 \ \right] + \begin{bmatrix} 4 & 1 & 0 \ -2 & 3 & -2 \ \right] = \begin{bmatrix} 0 + 4 & 3 + 1 & -5 + 0 \ 1 - 2 & 2 + 3 & 6 - 2 \ \right] = \begin{bmatrix} 4 & 4 & -5 \ -1 & 5 & 4 \ \right] \]

Check dimensions for matrix multiplication

Matrix C is 3x2 and the result of (A + B) is 2x3. To multiply matrices, the number of columns in the first matrix should equal the number of rows in the second matrix. Since (A + B) is 2x3 and C is 3x2, the multiplication C(A + B) is not defined.

Key concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra. To add two matrices, they must have the same dimensions, meaning they must have the same number of rows and columns. This is known as dimension compatibility.
In the given exercise, matrices A and B both have dimensions 2x3. Since their dimensions match, they can be added together element-wise, which means corresponding elements from each matrix are added together. For example, the element in the first row and first column of matrix A is added to the element in the first row and first column of matrix B.
The addition of matrices A and B is calculated as follows:
\[ A + B = \begin{bmatrix} 0 & 3 & -5 \ 1 & 2 & 6 \right] + \begin{bmatrix} 4 & 1 & 0 \ -2 & 3 & -2 \right] = \begin{bmatrix} 4 & 4 & -5 \ -1 & 5 & 4 \right] \]
In this resulting matrix, each element is the sum of the corresponding elements of A and B.

Matrix Multiplication

Matrix multiplication is another fundamental operation, but it is slightly more complex than addition. For two matrices to be multiplied, we need to check for dimension compatibility.
This means the number of columns in the first matrix (Matrix A) must be equal to the number of rows in the second matrix (Matrix B).
In the given exercise, we are asked to check if the expression \(C(A + B)\) is defined.
To multiply matrices, follow these steps:

• Calculate the dot product of rows of the first matrix with columns of the second matrix.
• Summing up the products of the corresponding elements forms the elements of the resulting matrix.

However, in this problem, Matrix C is a 3x2 matrix and \(A + B\) is a 2x3 matrix.
Since the number of columns in \(A + B\) is not equal to the number of rows in C, the multiplication of \(C(A + B)\) is not defined.
Always ensure dimension compatibility before attempting to multiply matrices.

Dimension Compatibility

Dimension compatibility is a key consideration in matrix operations.
It defines whether certain operations can be performed on given matrices.
For matrix addition, the matrices involved must have the exact same dimensions.
This ensures each element lines up correctly for the addition.
For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
This compatibility allows for the calculation of the dot product.
In the provided example:

• Matrix A is 2x3.
• Matrix B is 2x3.
• Matrix C is 3x2.

When we add Matrix A and Matrix B, the resulting matrix \(A + B\) is 2x3.
To multiply Matrix C by \(A + B\), we need the number of columns in \(A + B\) (which is 3) to match the number of rows in Matrix C (which is also 3).
But here, the resulting dimensions are \(C(A + B)\) will be 3x2 (from C) by 2x3 (from \(A + B\)), this operation is not compatible.
Understanding these rules of dimension compatibility is crucial in matrix operations to avoid undefined expressions.