Exercise Let ` A` be an invertible matrix.
Find

[ | A |
B |
] = [ | I |
O |
] [ | A |
O |
] [ | I |
Y |
] |

C |
D |
X |
I |
O |
S |
O |
I |

Then find out the condition for the matrix to be invertible.

Exercise A block upper triangular matrix is a partitioned matrix similar to the usual upper triangular matrix, except the entries are matrices instead of numbers, and the diagonal blocks are square matrices.

[ | A_{1} |
# |
. . | # |
], A_{1}, A_{2}, ..., are square matrices.A_{k} |

O |
A_{2} |
. . | # |
||

: | : | : | |||

O |
O |
. . | A_{k} |

What does the multiplication of two block upper triangular matrices look like? What about the inverse of a block upper triangular matrix? What about the similar problems for block lower triangular matrix and block diagonal matrix.