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Linear Algebra

Homework 2 - Gaussian elimination and invertibility

Due 2021-09-07.

  1. Show that for $c \neq 0$ \(S(i,c) S(i,1/c) = I.\) Is $S(i,c)$ invertible if $c = 0$?

  2. We already saw that if $A$ and $B$ are invertible then so is $AB$. Use induction to show: if $A_1,\ldots,A_t$ are all invertible, then the product $A_1 \cdots A_t$ are invertible.

  3. Prove that the only invertible matrices in reduced row echelon form are the identity matrices. Hint: look at the number of $0$ rows in a square reduced row echelon matrix. What can you say about the number of pivots? How are you going to squeeze all those pivots in?

  4. Use Gaussian elimination to determine if the following matrix has an inverse. Compute the inverse if it exists. \(\begin{pmatrix} 1 & -2 & 0 & 1 \\ 0 & -1 & 1 & 1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 1 & 0 \end{pmatrix}\)

  5. Use Gaussian elimination to determine if the following matrix has an inverse. Compute the inverse if it exists. \(\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}\)