Linear Algebra – Proving Upper Triangular Matrix Invertibility

linear algebramatrices

Prove that an upper triangular matrix is invertible if and only if every diagonal entry is non-zero.
I have proved that if every diagonal entry is non-zero, then the matrix is invertible by showing we can row reduce the matrix to an identity matrix. But how do I prove the only if part?

Best Answer

If $A$ is an $n\times n$ triangular matrix, consider the system of equations $$A\mathbf x=\mathbf 0$$

If last $0$ in the main diagonal is at position $j$, you can solve for $x_n$, $x_{n-1}$,...,$x_{j+1}$. But what happens with $x_j$? Must it be $0$?

But if $A$ had an inverse we would have $$A^{-1}A\mathbf x=\mathbf x=\mathbf 0$$

Can you complete the reasoning?

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