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" $$ A=\left[\begin{array}{rrr} 0 & 3 & -5 \\ 1 & 2 & 6 \end{array}\right] \quad B=\left[\begin{array}{rrr} 4 & 1 & 0 \\ -2 & 3 & -2 \end{array}\right] \quad C=\left[\begin{array}{rr} 4 & 1 \\ 6 & 2 \\ -2 & 3 \end{array}\right] $$ BA

Short answer

Not defined

Step-by-step solution

Determine Dimensions of Matrices

Matrix B has dimensions 2x3 (2 rows and 3 columns), and matrix A has dimensions 2x3. For the product BA to be defined, the number of columns in B must equal the number of rows in A.

Check Dimensional Compatibility

In this case, Matrix B has 3 columns, and Matrix A has 2 rows. Since the number of columns in B (3) does not match the number of rows in A (2), the product BA is not defined.

Key concepts

Matrix Dimensions

When working with matrices, understanding their dimensions is crucial. A matrix's dimensions are given by the number of rows and columns it has. For instance, if a matrix has 2 rows and 3 columns, its dimensions are expressed as 2x3. Knowing the dimensions helps us determine various matrix operations, such as addition, subtraction, and multiplication.

• Rows: The number of horizontal lines of elements.
• Columns: The number of vertical lines of elements.

For example, in our exercise, Matrix B has dimensions 2x3 (2 rows and 3 columns), and Matrix A also has dimensions 2x3. Remember, correctly identifying these dimensions is the first step in performing any matrix operation.

Matrix Multiplication Compatibility

Matrix multiplication is not as straightforward as element-wise multiplication. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. This is known as the 'inner dimension' rule. If this condition isn't met, the multiplication is undefined.

For example, in our exercise step-by-step solution:

• Matrix B has 3 columns and Matrix A has 2 rows.
• The number of columns in B (3) does not match the number of rows in A (2).
• Therefore, the product BA is undefined.

This is an essential concept to grasp because it saves time by identifying incompatible matrix operations quickly.

Undefined Matrix Operations

Sometimes, matrix operations are simply not possible. When this happens, the result is labelled as 'not defined'. The most common reasons for undefined matrix operations include:

• Non-matching dimensions: As we saw with the exercise, if the number of columns in one matrix does not match the number of rows in another, multiplication is not possible.
• Incompatible operations: Some matrix operations require specific conditions, like adding or subtracting matrices that must be of the same dimensions.

Understanding why operations are undefined helps in properly applying matrix operations and avoiding errors. For our exercise, since the number of columns in B (3) does not agree with the number of rows in A (2), the operation BA is not defined. Always check dimensions first to ensure the operation is feasible.