Question

For the given matrix, which of the statements given below is/are true?

\[ \begin{bmatrix} 3 & 1 & 5 \\ 2 & -1 & 0 \\ 5 & 2 & 9 \end{bmatrix} \]

• A. The matrix is full rank
• B. The matrix is invertible
• C. The matrix has a determinant
• D. The matrix has a transpose

Hint:

Compute the determinant first; if it is zero the matrix cannot be full rank or invertible, but it always has a determinant and a transpose.

Answer 1

The Correct Option is C, D

Step 1: Understanding the Question:
We are asked which general statements about a given $3\times3$ matrix are true. Two of the statements ("full rank," "invertible") depend on whether the matrix is singular, while the other two ("has a determinant," "has a transpose") are properties that hold for any matrix at all. Let's check singularity by looking at the rows directly instead of jumping straight to the cofactor formula.

Step 2: Key Formula or Approach:
A square matrix is full rank and invertible only if its rows are linearly independent, that is, no row can be written as a combination of the others. If such a combination exists, the matrix is singular, its determinant is 0, and it is neither full rank nor invertible, though it still always has a determinant (namely 0) and always has a transpose.

Step 3: Detailed Explanation:
The matrix is $\begin{bmatrix}3 & 1 & 5\\ 2 & -1 & 0\\ 5 & 2 & 9\end{bmatrix}$, with rows $R_1=(3,1,5)$, $R_2=(2,-1,0)$, $R_3=(5,2,9)$.
Adding the first two rows gives $R_1+R_2=(5,0,5)$, which is close to $R_3=(5,2,9)$ but not equal, so the dependency is not the simple sum; we confirm singularity directly through the determinant instead:
\[ \det = 3\big[(-1)(9)-(0)(2)\big] - 1\big[(2)(9)-(0)(5)\big] + 5\big[(2)(2)-(-1)(5)\big] = 3(-9)-1(18)+5(9) = -27-18+45 = 0 \]
A zero determinant confirms the rows are linearly dependent (some real combination of two rows reproduces the third), so the rank is less than 3.
Since rank $<3$ for a $3\times3$ matrix, the matrix is NOT full rank and NOT invertible: statements (A) and (B) are false.
Regardless of this, the determinant (here 0) still exists as a defined number, and the transpose (rows and columns swapped) still exists, for any matrix at all: statements (C) and (D) are always true.

Step 4: Final Answer:
The correct statements are (C) and (D), so the answer is options 3 and 4.