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

Short answer

Not defined.

Step-by-step solution

Determine the dimensions of matrices

Matrix A is a 2x3 matrix. Matrix C is a 3x2 matrix. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Therefore, the multiplication of C and A is defined if the number of columns in C (which is 2) equals the number of rows in A (which is 2).

Confirm if multiplication is possible

Since the number of columns in matrix C (3 columns) does not equal the number of rows in matrix A (2 rows), the multiplication CA is not defined.

Key concepts

Matrix Dimensions

Understanding matrix dimensions is crucial for matrix operations. A matrix is essentially a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are described by the count of its rows and columns. For instance, a matrix with 2 rows and 3 columns is referred to as a 2x3 matrix.

In our example, matrix A is a 2x3 matrix, meaning it has 2 rows and 3 columns. On the other hand, matrix C is a 3x2 matrix, indicating it consists of 3 rows and 2 columns.

Knowing the dimensions helps in determining if specific matrix operations, like multiplication, can be performed.

Matrix Product

Matrix product (or matrix multiplication) involves producing a third matrix from two input matrices. Each element of the resulting matrix is calculated by taking the dot product of corresponding row and column pairs of the input matrices.

For matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Suppose you have matrix D (m x n) and matrix E (n x p); the product matrix F will be of dimension (m x p).

For example, let matrix A be \([a_{ij}]\) with dimensions 2x3 and matrix B be \([b_{ij}]\) with dimensions 3x2. The product matrix C = AB will have the dimensions 2x2. Each element in C is obtained by calculating the dot product of rows of A and columns of B.

Conformability

Conformability in matrices indicates the condition under which matrix multiplication is possible. Matrices are conformable for multiplication if the number of columns in the first matrix matches the number of rows in the second matrix.

In simple terms, understand it as matching the inside dimensions of the paired matrices such as (a x b) and (b x c), where b's match indicates conformability.

Referring to our original exercise, matrix A is a 2x3 matrix, and matrix C is a 3x2 matrix. Multiplication CA requires the columns of C (which is 2) equal the rows of A (which is 2). However, since the columns in C (3) do not equal the rows in A (2), the multiplication CA is not defined. This is a prime example to clarify the concept of conformability in matrix multiplication.