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: Write down the matrix and recall what each term means.
\[ M = \begin{bmatrix} 3 & 1 & 5 \\ 2 & -1 & 0 \\ 5 & 2 & 9 \end{bmatrix} \]
"Full rank" means the number of linearly independent rows (or columns) equals the matrix size, here 3. "Invertible" means a matrix \(M^{-1}\) exists such that \(MM^{-1}=I\); a square matrix is invertible if and only if it is full rank, which happens if and only if its determinant is non-zero. "Has a determinant" and "has a transpose" are properties every square matrix possesses, regardless of its numerical value.

Step 2: Compute the determinant by cofactor expansion along the first row.
\[ \det(M) = 3\begin{vmatrix}-1 & 0 \\ 2 & 9\end{vmatrix} - 1\begin{vmatrix}2 & 0 \\ 5 & 9\end{vmatrix} + 5\begin{vmatrix}2 & -1 \\ 5 & 2\end{vmatrix} \]
\[ = 3\left[(-1)(9)-(0)(2)\right] - 1\left[(2)(9)-(0)(5)\right] + 5\left[(2)(2)-(-1)(5)\right] \]
\[ = 3(-9) - 1(18) + 5(9) = -27 - 18 + 45 = 0 \]

Step 3: Decide the truth of each statement.
Since \(\det(M)=0\), the matrix is singular, so it is NOT full rank (its rank is at most 2, because one row can be written as a combination of the other two) and it is NOT invertible (a zero determinant means \(M^{-1}\) does not exist). So statements (A) and (B) are false.
"The matrix has a determinant" is true regardless of the numerical value; every square matrix has a determinant, and here that value happens to be 0. So statement (C) is true.
"The matrix has a transpose" is also always true; any matrix, of any size or rank, has a transpose obtained by swapping rows and columns. So statement (D) is true.

Step 4: Sanity check the singular rank.
Row reduce: \(R_3 \to R_3 - R_1\) gives \([2,\,1,\,4]\); since the determinant already came out to 0, the three rows cannot all be independent, confirming the matrix cannot be full rank.

Final Answer:
The true statements are (C) "the matrix has a determinant" and (D) "the matrix has a transpose".
\[ \boxed{\text{(C) and (D)}} \]