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.