Exercise Prove that

[ | a |
b |
] |

c |
d |

is invertible ⇔ `ad` - `bc` ≠ 0.

Answer Denote the matrix by ` A`.

In the lecture, we have already shown that `ad` - `bc` ≠ 0 is necessary for ` A` to be invertible.

Conversely, assume `ad` - `bc` ≠ 0. Then either `a` ≠ 0 or `c` ≠ 0. If `a` ≠ 0, then the operation -`a`^{-1}`c`[row 1] + [row 2] on ` A` gives us

= [R |
a |
b |
] |

0 | d - a^{-1}cb |

Since `a` ≠ 0 and `d` - `a`^{-1}`cb` = `a`^{-1}(`ad` - `bc`) ≠ 0, all rows and columns of ` R` are pivot. By this criterion,

= [R |
c |
d |
] |

0 | b - c^{-1}ad |

Since `c` ≠ 0 and `b` - `c`^{-1}`ad` = -`c`^{-1}(`ad` - `BC`) ≠ 0, all rows and columns of ` R` are pivot. By this criterion,