Finding the Inverse of a Matrix?

Question

If I wanted to find the inverse of...

[5 2]

[-1 a]

(That's a 2x2 matrix)

How can I do it without a calculator?

Choosing best answer as always.

Thanks!

Answer 1

=1/5a+2[a -2; 1 5]

Answer 2

First step: Get the cofactor matrix, followed by the adjoint matrix, and finally calculate the determinant.

The inverse of the matric is then

M(adj)/M(determ).

See if  you can find something on the Internet or go to the library.

It is awkward to demonstrate it here because of the lack of math symbols on the key board

Answer 3

If it is non-singular matrix (regular matrix) then it's inverse will exist otherwise NOT.

Inverse matrix formula =

(Adjoint Matrix) / Determinant of original matrix

Calculate its determinant. Here 'a' is unknown, so therefore we can still write the determinant ,D = 5a + 2.

Now calculate its adjoint.

Step 1 : Form the cofactor matrix by calculating its minors ( To calculate the minor for an element we have to use the elements that do not fall in the same row and column of the minor element, i.e, take the determinant of the other remaining set of numbers and multiply by (-1) ^ (i + j) ) and then write it in place of the original element, this will form cofactor matrix.

Here,

Minor of element '5' will be a * (-1) ^ (1 + 1) = a, replace '5' by a.

Minor of element '2' will be -1 * (-1) ^ (1 + 2) = 1, replace 2 by 1

Minor of element '-1' will be 2 * (-1) ^ (2 + 1) = -2, replace -1 by 2

Minor of element 'a' will be 5 * (-1) ^ (2 + 2) = 5,  replace a by 5

Now the cofactor matrix is

[a 1]

[-2 5]

Step 2

Now to get the adjoint of this matrix, take the transpose of it, i.e., make colomns into rows and rows into colomns.

We will get

[a -2]

[1 5]

now multiply the above matrix by 1 / D = 1 / 5a + 2, we get the inverse matrix which is

[a/(5a + 2),  -2/(5a+2)]

[1/(5a+2), 5/(5a+2)]

Answer 4

find the determinant first

[5 2]   = [a b]

[-1 A]     [c d]

where a = 5, b = 2, c = -1, d = A

determinant is ad - bc

= 5A - 2(-1) = 5A+2

the inverse of any 2x2 matrix is 1/determinant x matrix T

which is

[d -b]

[-c a]

so the inverse to your matrix is

[A -2]

[1 5 ]   x 1/(5A +2)

=[A/(5A+2)  -2/(5A+2)]

  [1/(5A+2 )       5/(5A+2]

          

edit: this is 100% the right way to do do it for 2x2 matrices, the answerer below is suggesting a method for 3x3 matrices and higher.