Thread: Solving a Matrix using its Determinant

1. Solving a Matrix using its Determinant

Hi all, I'm practicing for a mid-semester linear algebra exam and I'm stuck on this question:

Let Ax=b where A = $\displaystyle \begin{bmatrix} 2 & 1 & 0 \\ -3 & 0 & 1\\ 0 & 1 & 2 \end{bmatrix}$ , det A = 4 and b = $\displaystyle \begin{bmatrix} 7 \\ -8 \\ -3 \end{bmatrix}$. What does x2 equal?

I know I could just use row reduction to solve for x2 but I would like to know how to use the determinant to make the problem easier.

Thanks,

2. Re: Solving a Matrix using its Determinant

Originally Posted by anguished
Hi all, I'm practicing for a mid-semester linear algebra exam and I'm stuck on this question:

Let Ax=b where A = [2, 1, 0; -3, 0, 1; 0, 1, 2], det A = 4 and b = [7, -8, -3]. What does x2 equal?

I know I could just use row reduction to solve for x2 but I would like to know how to use the determinant to make the problem easier.

Thanks,
Are you sure b doesn't equal [7; -8; -3]?

3. Re: Solving a Matrix using its Determinant

Yes, that's what I meant sorry. I should have used Latex instead. :P Fixed now.

4. Re: Solving a Matrix using its Determinant

Originally Posted by anguished
Hi all, I'm practicing for a mid-semester linear algebra exam and I'm stuck on this question:

Let Ax=b where A = $\displaystyle \begin{bmatrix} 2 & 1 & 0 \\ -3 & 0 & 1\\ 0 & 1 & 2 \end{bmatrix}$ , det A = 4 and b = $\displaystyle \begin{bmatrix} 7 \\ -8 \\ -3 \end{bmatrix}$. What does x2 equal?

I know I could just use row reduction to solve for x2 but I would like to know how to use the determinant to make the problem easier.

Thanks,
You need to realise that if $\displaystyle \displaystyle \mathbf{A}\mathbf{x} = \mathbf{b}$, then $\displaystyle \displaystyle \mathbf{x} = \mathbf{A}^{-1}\mathbf{b}$.

You can use the determinant to help evaluate $\displaystyle \displaystyle \mathbf{A}^{-1}$.