Answer to Question #195408 in Linear Algebra for Tshilidzi

Question

Suppose A, B, C and D are matrices with the following sizes A(5x2), B(4x2), C(4X5), and D(4x5)

Determine each of the following case whether a product is defined. If it is so then give the size of the resulting matrix

a. DC

b. -CA+B

c. CD-D

Answer 1

We can oly multiply two matrices if their dimensions are compatible, which means the number of columns in the first matrix is the same as the number of rows in the second matrix.

If $A=[a_{ij}]$ is an $m\times n$ matrix and $B=[b_{ij}]$  is an $n\times p$ matrix, the product $AB$ is an $m\times p$ matrix.

The operation of addition of two matrices is only defined when both matrices have the same dimensions. If $A$ and $B$ are both $m\times n,$ then the sum $C=A+B$ is also $m\times n.$

We can also compute the difference $D=A-B$ by summing $A$ and $(-1)B$

$$D=A-B=A+(-1)B$$

a. Since $D$ is an $4\times 5$ matrix and $C$ is an $4\times 5$ matrix, then the matrix product $DC$ is not defined.

b. Since $C$ is an $4\times 5$ matrix and $A$ is an $5\times 2$ matrix, then the matrix product $-CA$ is defined and is $4\times 2$ matrix.

Since $B$ is $4\times 2$ matrix , then the operation of addition of two matrices $-CA+B$ is defined, and the sum $-CA+B$ is $4\times 2$ matrix.

c. Since $C$ is an $4\times 5$ matrix and $D$ is an $4\times 5$ matrix, then the matrix product $CD$ is not defined.