Consider the following statements about the matrix $M = \begin{vmatrix} 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 \end{vmatrix}$ Statement-I : The inverse of $M$ does not exist. Statement-II : $M$ is non-singular. Which one of the following is correct in respect of the above statements?
Statement-I is correct but Statement-II is not correct
To determine the correctness of Statement-I and Statement-II, we need to analyze them with respect to the properties of matrix $M$.
Statement-I: "The inverse of $M$ does not exist."
The inverse of a square matrix exists if and only if the matrix is nonsingular. A matrix is nonsingular if its determinant is non-zero. Hence, we will first compute the determinant of matrix $M$.
| \(M = \begin{vmatrix} 71 & 23 & 48 \\ 57 & 28 & 29 \\ 65 & 17 & 48 \end{vmatrix}\) |
The determinant of matrix \(M\) is computed as follows:
- Expand along the first row: \(\text{det}(M) = 71 \begin{vmatrix} 28 & 29 \\ 17 & 48 \end{vmatrix} - 23 \begin{vmatrix} 57 & 29 \\ 65 & 48 \end{vmatrix} + 48 \begin{vmatrix} 57 & 28 \\ 65 & 17 \end{vmatrix}\)
- Calculate each of the 2x2 determinants:
- \(\begin{vmatrix} 28 & 29 \\ 17 & 48 \end{vmatrix} = (28 \times 48) - (29 \times 17) = 1344 - 493 = 851\)
- \(\begin{vmatrix} 57 & 29 \\ 65 & 48 \end{vmatrix} = (57 \times 48) - (29 \times 65) = 2736 - 1885 = 851\)
- \(\begin{vmatrix} 57 & 28 \\ 65 & 17 \end{vmatrix} = (57 \times 17) - (28 \times 65) = 969 - 1820 = -851\)
- Substitute these values back: \(\text{det}(M) = 71 \times 851 - 23 \times 851 + 48 \times (-851)\)
- Factor out \(851\): \(\text{det}(M) = 851(71 - 23 - 48) = 851 \times 0 = 0\)
The determinant of \(M\) is 0, which means that \(M\) is a singular matrix. Hence, the inverse of \(M\) does not exist, making Statement-I correct.
Statement-II: "M is non-singular."
However, from our calculation, it is clear that \(M\) is singular, as its determinant is 0. Therefore, Statement-II is not correct.
Based on the analysis, the correct answer is:
Statement-I is correct but Statement-II is not correct
Important Questions from Matrices and Determinants
What is the value of the determinant of the inverse of the matrix $\begin{bmatrix} -4 & -5 \\ 2 & 2 \end{bmatrix}$?
In obtaining the solution of the system of equations $x + y + z = 7$, $x + 2y+3z = 16$ and $x + 3y+4z = 22$ by Cramer's rule, the value of $y$ is obtained by dividing $D$ by $D_2$, where
$D = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 4 \end{vmatrix}$
What is the value of the determinant $D_2$?
Consider the following in respect of non-singular matrices $A$ and $B$ :
I. $(AB)^{-1} = A^{-1}B^{-1}$
II. $(BA)(AB)^{-1} = I$, where $I$ is the identity matrix
III. $(AB)^T = A^T B^T$
How many of the above are correct?
Consider the following statements :
Statement-I :
If $X$ is an $n \times n$ matrix, then $\det(mX) = m^n \det(X)$, where $m$ is a scalar.
Statement-II :
If $Y$ is a matrix obtained from $X$ by multiplying any row or column by a scalar $m$, then $\det(Y) = m\det(X)$.
Which one of the following is correct in respect of the above statements?
- Let $f(x) = \begin{vmatrix} 3x^2 & \cos x & -\sin x \\ 6 & -1 & 0 \\ q & q^2 & q^3 \end{vmatrix}$ where q is any constant, then what is $\frac{d^2}{dx^2}(f(x))$ at $x = 0$ equal to ?
