I saw in lecture of Linear algebra, while doing problem sessions two things but with no justification from teacher.

Consider $A,B$ as $2n\times 2n$ matrices over a field $F$. While doing multiplication of them suppose we partition them in a suitable way as

$$A=\begin{bmatrix} A_1 & A_2 \\ A_3 & A_4\end{bmatrix}, B=\begin{bmatrix} B_1 & B_2 \\ B_3 & B_4\end{bmatrix}$$ where all $A_i$'s and $B_i$'s are $n\times n$ matrices. Then in the calculations it was written, like as in usual matrix multiplication

$$AB=\begin{bmatrix}A_1B_1 + A_2B_3 & A_1B_2 +A_2B_4\\ \cdots & \cdots\end{bmatrix}.$$

Q.1 How can we justify in elementary way that this multiplication is same as the usual multiplication without partitioning into blocks? In general, is there neat way to do matrix multiplications by partitioning in some specific type of blocks?

Q.2 Turn to determinants from matrices; suppose we want to find determinant of $A$ written in above form. Then is it always equal to $\det(A_1)\det(A_4)-\det(A_2)\det(A_3)$?

Q.3 Which book on matrices or Linear algebra describes these very elementary things with details of theory (theorems and proofs) as well as examples?

  • 2
    $\begingroup$ recommend you just do some 4 by 4 examples. The blocks will be 2 by 2. The determinant cannot be that simple in general, as products that mix terms from all the blocks are present. $\endgroup$ – Will Jagy Jan 18 '17 at 5:21
  • $\begingroup$ These sort of objects are called Block Matrices and you can find some of the results under that header. The idea in Q2 for example turn out not to hold some similar results do, for example if the block matrix is "block-upper triangular", $A_3 = 0$, then the determinant becomes the product of the determinants of the blocks forming the diagonal $\det A_1 \det A_4$ similar to the scalar case. That particular one I found in Shilov's Linear Algebra p.23 which isn't about block matrices overall though. $\endgroup$ – Squid Jan 18 '17 at 5:23
  • $\begingroup$ Will: OK, I will work out as you said. Can you tell a book for expansion of determinants in different ways? $\endgroup$ – Beginner Jan 18 '17 at 5:24
  • $\begingroup$ I don't know any books for beginners. Horn and Johnson is good and includes some block decompositions, but it would not be easy reading. $\endgroup$ – Will Jagy Jan 18 '17 at 5:30
  • $\begingroup$ If you simply put determinant block matrix into Google, you will find several basic results related to your question 2. It only works under some conditions, the simplest is probably the case when one od the four blocks is zero. Several other conditions can be found in Silvester: Determinants of Block Matrices. Wikipedia article Block matrix might be worth checking too. $\endgroup$ – Martin Sleziak Jan 18 '17 at 6:19

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