Determine if you can multiply the two matrices, then find the dimensions of the resulting matrix.
The product $X Y$, if $X=\left[\begin{array}{lll}5 & 2 & 7\end{array}\right]$ and $Y=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$.
A. Yes, the solution is a $1 x 1$ matrix.
B. Yes, the solution is a $3 x 1$ matrix.
C. Yes, the solution is a $3 x 3$ matrix.
D. No, you cannot multiply the matrices.
Answer (2)
Check if the number of columns of the first matrix equals the number of rows of the second matrix.
Determine the dimensions of the resulting matrix.
The product X Y is a 1 × 1 matrix.
The final answer is a 1 × 1 matrix: 1 × 1 .
Explanation
Understanding the Problem We are given two matrices, X and Y , and we want to determine if we can multiply them and, if so, find the dimensions of the resulting matrix. Matrix X is a 1 × 3 matrix, and matrix Y is a 3 × 1 matrix.
Checking Matrix Multiplication Condition To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. In this case, X has 3 columns and Y has 3 rows. Since these numbers are equal, we can multiply the matrices.
Determining the Dimensions of the Resulting Matrix When we multiply a 1 × 3 matrix by a 3 × 1 matrix, the resulting matrix has dimensions 1 × 1 .
Final Answer Therefore, the product X Y is a 1 × 1 matrix.
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 rotations, scaling, and translations of objects in 3D space. Multiplying a matrix representing a transformation by a matrix representing an object's coordinates results in a new matrix representing the transformed object. This allows for efficient manipulation and rendering of complex scenes.
You can multiply the matrices X and Y , resulting in a 1 × 1 matrix. Therefore, the correct option is A. The multiplication is defined since the number of columns in X equals the number of rows in Y .
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