Solving for |M| |adjM| of Scalar Matrix M
We are given a scalar matrix M and need to find the value of the expression $|M| |adjM|$.
Matrix M Definition
The given matrix is:
$ M = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} $
Step 1: Calculate the Determinant |M|
For a diagonal or scalar matrix, the determinant is the product of the diagonal elements.
- $|M| = 2 \times 2 \times 2$
- $|M| = 8$
Step 2: Understand the Adjoint Matrix Property
For any $n \times n$ matrix $M$, the determinant of its adjoint matrix is related to the determinant of $M$ by the formula:
$|adjM| = |M|^{n-1}$
In this case, $M$ is a $3 \times 3$ matrix, so $n=3$.
- $|adjM| = |M|^{3-1}$
- $|adjM| = |M|^2$
Step 3: Calculate |adjM|
Using the value of $|M|$ calculated in Step 1:
- $|adjM| = (8)^2$
- $|adjM| = 64$
Step 4: Calculate the Final Expression |M| |adjM|
Now, multiply the determinant $|M|$ by the determinant of the adjoint $|adjM|$:
- $|M| |adjM| = |M| \times |adjM|$
- $|M| |adjM| = 8 \times 64$
- $|M| |adjM| = 512$
Alternatively, we can directly calculate $|M|^3$ since $|M| |adjM| = |M| \cdot |M|^{n-1} = |M|^n$. For $n=3$, this is $|M|^3$.
- $|M|^3 = 8^3$
- $|M|^3 = 512$
Conclusion
The value of $|M| |adjM|$ for the given matrix $M$ is 512.
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?
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?
