What is the value of the determinant of the inverse of the matrix $\begin{bmatrix} -4 & -5 \\ 2 & 2 \end{bmatrix}$?
Solving for the Determinant of an Inverse Matrix
This problem asks us to find the value of the determinant of the inverse of a given 2x2 matrix. Let the matrix be denoted by A.
We need to find $det(A^{-1})$.
Key Concepts for Matrix Determinants
There are a couple of important properties of determinants that make this calculation simpler:
- The determinant of a 2x2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is calculated as $ad - bc$.
- A crucial property relating a matrix and its inverse is: $det(A^{-1}) = \frac{1}{det(A)}$. This means we don't need to calculate the inverse matrix itself to find its determinant.
Step 1: Calculate the Determinant of the Original Matrix (A)
First, let's find the determinant of the given matrix A using the formula $ad - bc$.
For matrix $A = \begin{bmatrix} -4 & -5 \\ 2 & 2 \end{bmatrix}$, we have $a = -4$, $b = -5$, $c = 2$, and $d = 2$.
Calculating the determinant:
Step 2: Use the Determinant Property to Find $det(A^{-1})$
Now, we apply the property $det(A^{-1}) = \frac{1}{det(A)}$.
Since we found that $det(A) = 2$, we can substitute this value:
Conclusion
The value of the determinant of the inverse of the matrix $\begin{bmatrix} -4 & -5 \\ 2 & 2 \end{bmatrix}$ is $\frac{1}{2}$.
Important Questions from Matrices and Determinants
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?
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?
- 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 ?
