# Thread: Determinant and inverse matrix

2. ## Re: Determinant and inverse matrix

How do i solve this question? I have been trying it for a hour and the answers i got didnt match any of the options... I already used A^-1=1/|A|*A^^ but the element c(1,4) is 0 to me dont know why...

3. ## Re: Determinant and inverse matrix

Originally Posted by kym
How do i solve this question? I have been trying it for a hour and the answers i got didnt match any of the options... I already used A^-1=1/|A|*A^^ but the element c(1,4) is 0 to me dont know why...
$A^{-1}_{1,4} = \dfrac{(-1)\left | \begin{pmatrix}2 &0 &-1 \\0 &1 &-4 \\1 &0 &0 \end{pmatrix}\right|}{|A|} = -\dfrac{1}{4}$

$\begin{pmatrix}2 &0 &-1 \\0 &1 &-4 \\1 &0 &0 \end{pmatrix}$ is the minor matrix of $A_{4,1}$

$(-1) = (-1)^{1+4}$ turns the minor into a cofactor

4. ## Re: Determinant and inverse matrix

Originally Posted by romsek
$A^{-1}_{1,4} = \dfrac{(-1)\left | \begin{pmatrix}2 &0 &-1 \\0 &1 &-4 \\1 &0 &0 \end{pmatrix}\right|}{|A|} = -\dfrac{1}{4}$

$\begin{pmatrix}2 &0 &-1 \\0 &1 &-4 \\1 &0 &0 \end{pmatrix}$ is the minor matrix of $A_{4,1}$

$(-1) = (-1)^{1+4}$ turns the minor into a cofactor
What formula is that? I thought that above the determinant of A i had to use the adjugate matrix of A and find the element {1,4}. How can you use just A4,1 to solve that if it isn't even a matrix to stand above the determinant? I'm so confused with determinants in general :/...

5. ## Re: Determinant and inverse matrix

Originally Posted by kym
What formula is that? I thought that above the determinant of A i had to use the adjugate matrix of A and find the element {1,4}. How can you use just A4,1 to solve that if it isn't even a matrix to stand above the determinant? I'm so confused with determinants in general :/...
think about the steps you need to find the inverse matrix

first you find the determinant of each element's minor matrix

then you apply the sign correction to convert the minor determinants to cofactors

then you divide by the determinant of the original matrix

then you transpose the elements

so for element (1,4) of the inverse we need to calculate element (4,1) of the cofactor matrix and divide by the determinant of the original matrix.

that's exactly what this does.

I didn't get any formula anywhere. I just understand how the inverse is computed using the cofactor method.

6. ## Re: Determinant and inverse matrix

Originally Posted by romsek
think about the steps you need to find the inverse matrix

first you find the determinant of each element's minor matrix

then you apply the sign correction to convert the minor determinants to cofactors

then you divide by the determinant of the original matrix

then you transpose the elements

so for element (1,4) of the inverse we need to calculate element (4,1) of the cofactor matrix and divide by the determinant of the original matrix.

that's exactly what this does.

I didn't get any formula anywhere. I just understand how the inverse is computed using the cofactor method.
Thanks a lot man! I don't knew we could use such methods to do so... I was guiding myself only by the formulas.

I'm having algebra exam in less than a week, if you dont mind, could you give some youtube channel or nice study material to prepare myself better? Would apreciate the help.