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.
