Question 6.1.11: Find det M for M=[a11 a12 b11 b12 a21 a22 b21 b22 0 0 c11 c1......
Find det M for
M=\begin{bmatrix}a_{11}&a_{12}&b_{11}&b_{12}\\a_{21}&a_{22}&b_{21}&b_{22}\\0&0&c_{11}&c_{12}\\0&0&c_{21}&c_{22}\end{bmatrix}.
It is natural to partition the 4 × 4 matrix M into four 2 × 2 blocks, one of which is zero:
M=\begin{bmatrix}A&B\\0&C\end{bmatrix}.
Let’s see whether we can express det \begin{bmatrix}A&B\\0&C\end{bmatrix} in terms of det A, det B, and detC.
Let’s find the patterns in M that may have a nonzero product.
Thus
det M = a_{11}a_{22}c_{11}c_{22} − a_{11}a_{22}c_{12}c_{21} − a_{12}a_{21}c_{11}c_{22} + a_{12}a_{21}c_{12}c_{21}
= a_{11}a_{22}(c_{11}c_{22} − c_{12}c_{21}) − a_{12}a_{21}(c_{11}c_{22} − c_{12}c_{21})
= (a_{11}a_{22} − a_{12}a_{21})(c_{11}c_{22} − c_{12}c_{21}) = (det A)(detC)..
In summary,
\det M = \det\begin{bmatrix}A&B\\0&C\end{bmatrix}=(\det A)(\det C).