Which of the answer choices has matrix multiplication defined?
[tex]\begin{array}{lccc}
A & B & C & D \\
{\left[\begin{array}{l}
1 \\
3 \\
5
\end{array}\right]} & {\left[\begin{array}{lll}
1 & 2 & 7 \\
6 & 8 & 0
\end{array}\right]} & {\left[\begin{array}{lll}
5 & 9 & 7
\end{array}\right]} & {\left[\begin{array}{ccc}
-1 & 2 & 3 \\
8 & -4 & 0 \\
6 & 7 & 1
\end{array}\right]}
\end{array}[/tex]
A. AD
B. DB
C. AB
Answer (2)
Matrix A is 3x1, B is 2x3, C is 1x3, and D is 3x3.
AD is not defined because the number of columns in A (1) is not equal to the number of rows in D (3).
DB is not defined because the number of columns in D (3) is not equal to the number of rows in B (2).
AB is not defined because the number of columns in A (1) is not equal to the number of rows in B (2).
Therefore, none of the given matrix multiplications are defined. N o n e
Explanation
Understanding Matrix Multiplication We are given four matrices A, B, C, and D, and we need to determine which of the matrix multiplications AD, DB, and AB is defined. Matrix multiplication is defined if the number of columns in the first matrix is equal to the number of rows in the second matrix.
Identifying Matrix Dimensions Matrix A is a 3x1 matrix, meaning it has 3 rows and 1 column. Matrix B is a 2x3 matrix, meaning it has 2 rows and 3 columns. Matrix C is a 1x3 matrix, meaning it has 1 row and 3 columns. Matrix D is a 3x3 matrix, meaning it has 3 rows and 3 columns.
Checking AD For matrix multiplication AD to be defined, the number of columns in A must equal the number of rows in D. A has 1 column and D has 3 rows. Since 1 e q 3 , AD is not defined.
Checking DB For matrix multiplication DB to be defined, the number of columns in D must equal the number of rows in B. D has 3 columns and B has 2 rows. Since 3 e q 2 , DB is not defined.
Checking AB For matrix multiplication AB to be defined, the number of columns in A must equal the number of rows in B. A has 1 column and B has 2 rows. Since 1 e q 2 , AB is not defined.
Conclusion Therefore, none of the given matrix multiplications are defined.
Examples
Matrix multiplication is a fundamental operation in computer graphics. For example, transformations like scaling, rotation, and translation of 3D objects in space can be represented using matrices. Checking if matrix multiplication is defined ensures that these transformations can be applied correctly. If the dimensions are incompatible, the transformation would be invalid, leading to errors in the rendered image.
None of the matrix multiplications AD, DB, or AB are defined due to incompatible dimensions. Matrix A cannot multiply with either matrix D or B. Therefore, the answer is that none of them are defined.
;
