Question

Let $A$ be $3 \times 3$, $B$ be $3 \times 2$ and $C$ be $3 \times 1$. Which one of the following products is not defined? ________.

• A. $C^{T}AB$
• B. $A^{T}AB$
• C. $(AB)^{T}C$
• D. $(AB)C$
• E. $B^{T}C$

Hint:

$(m \times n) \times (n \times p) \to m \times p$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
The product of two matrices, say P of dimension \(m \times n\) and Q of dimension \(p \times q\), is defined only if the number of columns in the first matrix (n) is equal to the number of rows in the second matrix (p). The resulting matrix PQ will have dimensions \(m \times q\). The transpose of a matrix \(M_{m \times n}\) is \(M^T_{n \times m}\).
Step 2: Key Formula or Approach
We are given the dimensions of the matrices:
- A is \(3 \times 3\)
- B is \(3 \times 2\)
- C is \(3 \times 1\)
From these, we can determine the dimensions of their transposes:
- \(A^T\) is \(3 \times 3\)
- \(B^T\) is \(2 \times 3\)
- \(C^T\) is \(1 \times 3\)
We will check the validity of each product option by matching the inner dimensions.
Step 3: Detailed Explanation
Let's check each option:
(A) \(C^T AB\):
- First, consider \(C^T A\): Dimensions are \((1 \times 3)\) and \((3 \times 3)\). The inner dimensions (3 and 3) match. The result \((C^T A)\) is a \(1 \times 3\) matrix.
- Now, consider \((C^T A)B\): Dimensions are \((1 \times 3)\) and \((3 \times 2)\). The inner dimensions (3 and 3) match. The product is defined.
(B) \(A^T AB\):
- First, consider \(A^T A\): Dimensions are \((3 \times 3)\) and \((3 \times 3)\). The inner dimensions (3 and 3) match. The result \((A^T A)\) is a \(3 \times 3\) matrix.
- Now, consider \((A^T A)B\): Dimensions are \((3 \times 3)\) and \((3 \times 2)\). The inner dimensions (3 and 3) match. The product is defined.
(C) \((AB)^T C\):
- First, consider AB: Dimensions are \((3 \times 3)\) and \((3 \times 2)\). The inner dimensions (3 and 3) match. The result (AB) is a \(3 \times 2\) matrix.
- The transpose \((AB)^T\) will have dimensions \(2 \times 3\).
- Now, consider \((AB)^T C\): Dimensions are \((2 \times 3)\) and \((3 \times 1)\). The inner dimensions (3 and 3) match. The product is defined.
(D) (AB)C:
- First, consider AB: Dimensions are \((3 \times 3)\) and \((3 \times 2)\). The inner dimensions (3 and 3) match. The result (AB) is a \(3 \times 2\) matrix.
- Now, consider (AB)C: The dimensions are \((3 \times 2)\) and \((3 \times 1)\). The number of columns of AB (2) does not match the number of rows of C (3). Therefore, this product is not defined.
(E) \(B^T C\):
- The dimensions are \((2 \times 3)\) and \((3 \times 1)\). The inner dimensions (3 and 3) match. The product is defined.
Step 4: Final Answer
The product that is not defined is (AB)C.