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Let $A$, $B$ be $n$ by $n$ matrices. Let $0$ be the all zero matrix of size $n$. Show that

$$\det\left[\begin{array}[cc]\\A& 0\\ 0&B\end{array}\right]=\det(A)\det(B)$$

What i tried

What i tried was first letting $A$ and $B$ be a simple 2 by 2 matrix and observing how the determinant relate to each other and then working my up. WHat i can deduce is that, if $A$ is singular than this will maake both sides of the equation $0$ hence proving the equation. Im currently working on the case when $A$ is not singular Could anyone explain. Thanks

ys wong
  • 2,065
  • What definition of the determinant do you use? – A.Γ. Nov 03 '15 at 06:03
  • Induction on the number of row operations needed to get the big matrix into upper-triangular form. – Christopher Carl Heckman Nov 03 '15 at 06:07
  • What i tried was first letting $A$ and $B$ be a simple 2 by 2 matrix by observing how the determinant relate to each other and then working my up. WHat i can deduce is that, if $A$ is singular than this will maake both sides of the equation $0$ hence proving the equation. Im still working on the case when $A$ is not singular – ys wong Nov 03 '15 at 06:09
  • http://math.stackexchange.com/questions/75293/determinant-of-a-block-lower-triangular-matrix –  Nov 03 '15 at 06:18

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