Which of the answer choices has matrix multiplication defined?
$\begin{array}{ccc}
A & B & C \\
{\left[\begin{array}{ll}
1 & 2 \\
3 & 4
\end{array}\right]} & {\left.\begin{array}{lll}
5 & 6 & 7 \\
8 & 9 & 0
\end{array}\right]} & D \\
{\left[\begin{array}{ccc}
-1 & 2 & 3 \\
8 & -4 & 0 \\
6 & 7 & 1
\end{array}\right]\left[\begin{array}{lll}
5 & 9 & 7
\end{array}\right]}
\end{array}$
A. $AC$
B. $CD$
C. $BA$
D. $AB$
Answer (1)
Matrix multiplication is defined if the number of columns of the first matrix equals the number of rows of the second matrix.
Check each option: A C , C D , B A , and A B .
A B is the only defined matrix multiplication since A is 2 × 2 and B is 2 × 3 .
Therefore, the answer is A B .
Explanation
Understanding Matrix Multiplication We are given four options of matrix multiplications: A C , C D , B A , and A B . We need to determine which of these multiplications is defined. Matrix multiplication is defined if the number of columns of the first matrix is equal to the number of rows of the second matrix.
Identifying Matrix Dimensions Matrix A is a 2 × 2 matrix. Matrix B is a 2 × 3 matrix. Matrix C is a 3 × 1 matrix.
Checking AC For A C to be defined, the number of columns of A must equal the number of rows of C . A is 2 × 2 and C is 3 × 1 . The number of columns of A is 2, and the number of rows of C is 3. Since 2 = 3 , A C is not defined.
Checking CD For C D to be defined, the number of columns of C must equal the number of rows of D . C is 3 × 1 and D is 3 × 3 . The number of columns of C is 1, and the number of rows of D is 3. Since 1 = 3 , C D is not defined.
Checking BA For B A to be defined, the number of columns of B must equal the number of rows of A . B is 2 × 3 and A is 2 × 2 . The number of columns of B is 3, and the number of rows of A is 2. Since 3 = 2 , B A is not defined.
Checking AB For A B to be defined, the number of columns of A must equal the number of rows of B . A is 2 × 2 and B is 2 × 3 . The number of columns of A is 2, and the number of rows of B is 2. Since 2 = 2 , A B is defined.
Conclusion Therefore, the only matrix multiplication that is defined is A B .
Examples
Matrix multiplication is used in various fields such as computer graphics, physics, and economics. For example, in computer graphics, matrices are used to represent transformations such as rotation, scaling, and translation of objects. Multiplying a matrix representing an object's coordinates by a transformation matrix applies the transformation to the object. In economics, matrix multiplication can be used to model the relationships between different sectors of an economy.
