HOW TO FIND THE INVERSE OF 2×2 AND 3×3 MATRIX USING SHORTCUT METHOD
INVERSE OF 2×2 AND 3×3 MATRIX
Hello and Welcome to this post ,Today we are going to discuss the shortest and easiest methods of finding the Inverse of 2×2 matrix and 3×3 Matrix.INVERSE OF 2×2 AND 3×3 MATRIX, shortcut to find inverse of 2 × 2 matrix,transpose 3x3,inverse of determinant,matrices and determinants, determinant calculator, crammer's rule.
Usually when we have to find the Inverse of any Matrix then we follow the following steps .
1 Check whether the determinant value of the given Matrix is Non Zero.
2 Find out the co-factors of all the elements of the Matrix.
3 Put these co-factors in co-factor Matrix.
4 Find the Ad joint of this matrix by taking the Transpose of a Matrix of the co-factor matrix.
5 Now Multiply Ad Joint of Matrix with the reciprocal of Determinant value of the given Matrix.
This Method is very confusing, Long and time Consuming. So Let us have a New, Easy and Shortcut Method .
Method For 2×2 Matrix
If we have to find the Inverse of 2×2 Matrix then Follows these steps.
1 Interchange the position of the elements which are a11 and a22 .
2 Change the Magnitude of the elements which are in position a12 and a21 .
3 Divide every elements of the given Matrix with its Determinant value.
Example
To find the Inverse of this matrix just interchange the position of elements a₁₁ and a₂₂ i.e Interchange the positions of elements 5 and -3 and in second step change the magnitude of the elements which are in positions a12 and a21 i.e. change the sign of 9 and 4.
Now divide each elements with determinants value of the matrix which is (5)(-3) - (9)(4) = -15 -36 = -51
So The Inverse of the given Matrix A will be
Then after interchanging the positions of 8 and 2 change the magnitude of 7 and -6 and divide every elements with its determinant value (8)( 2) - (7)*(-6) = 16+42 = 58
The Inverse of B is
After interchanging the position of -3 and -6 and changing the magnitude of -4 and -5 and at last dividing every elements with its determinant value (-3)×(-6) - (-4)×(-5) = 18 - 20 = -2
The Inverse of C is
This video Explains all about Inverse of 2×2 and 3×3 Matrix
Method for 3×3 Matrix
Ist of all Write the given Matrix in five columns by adding the 4th column as repetition of 1st column and 5th column as repetition of 2nd column, then
C₁ C2 C3 C4 C5
5 -1 4 5 -1
2 3 5 2 3
5 -2 6 5 -2
Now Expanding this Matrix to 5×5 Matrix by adding 4th Row as repetition of 1st Rows and adding 5 Row as repetition of 2nd column as what we received in last step.
R₁ 5 -1 4 5 -1
R₂ 2 3 5 2 3
R₃ 5 -2 6 5 -2
R₄ 5 -1 4 5 -1
R5 2 3 5 2 3
Now to find the Inverse of the given Matrix ,we have to find the cofactor of every elements
1 Find the co-factor of 1st element of Row 1 i. e. 5, determinant value of the Matrix (RED below ) obtained by eliminating the 1st Row and 1st Column which will be (3×6)-{(5)×(-2)} = 28,write these co-factor value in 1st column of 1st Row. (we are evaluating co-factors row wise and writing Column wise)
R₁ 5 -1 4 5 -1
R₂ 2 3 5 2 3
R₃ 5 -2 6 5 -2
R₄ 5 -1 4 5 -1
R5 2 3 5 2 3
2 Now Find the co-factor of 2nd element of 1st Row i. e. -1,which is equal to determinant value of the Matrix (RED below) obtained by eliminating the 1st Row and 2nd Column which will be 5*5-(2)*(6) =13,write this co-factor value in 2nd Row of 1st column .(we are evaluating co-factors row wise and writing Column wise)
R₁ 5 -1 4 5 -1
R₂ 2 3 5 2 3
R₃ 5 -2 6 5 -2
R₄ 5 -1 4 5 -1
R5 2 3 5 2 3
3 Now Find the co-factor of 3rd element of 1st Row i.e. 4, which is equal to determinant value of the Matrix (RED below ) obtained by eliminating the 1st Row and 3rd Column which will be 2*(-2)-(3)*(5) = -19,write this co-factor value in 3rd Row of 1st column.(we are evaluating co factors row wise and writing Column wise)
R₁ 5 -1 4 5 -1
R₂ 2 3 5 2 3
R₃ 5 -2 6 5 -2
R₄ 5 -1 4 5 -1
R5 2 3 5 2 3
4 Now Find the co-factor of 1st element of 2nd Row i. e. 2, which is equal to determinant value of the Matrix (RED below ) obtained by eliminating the 2nd Row and 1st Column which will be -2*(4)-(6)*(-1) = -2,write this co-factor value in 2nd Column of 1st Row .(we are evaluating co factors row wise and writing Column wise) .
R₁ 5 -1 4 5 -1
R₂ 2 3 5 2 3
R₃ 5 -2 6 5 -2
R₄ 5 -1 4 5 -1
R5 2 3 5 2 3
5 Now Find the co-factor of 2nd element of 2nd Row i.e 3, which is equal to determinant value of the matrix (RED below ) obtained by eliminating the 2nd Row and 2nd column which will be 6*(5)-(5)*(4) = 10,write this co-factor value in 2nd Row of 2nd column
R₁ 5 -1 4 5 -1
R₂ 2 3 5 2 3
R₃ 5 -2 6 5 -2
R₄ 5 -1 4 5 -1
R5 2 3 5 2 3
6 Find the co-factor of 3rd element of 2nd Row i. e.5, which is equal to determinant value of the Matrix (RED ) obtained by eliminating the 2nd Row and 2nd Column which will be 5× (-1)-(-2) × (5) = 5,write this co-factor value this 2nd Column of 3rd Row, write this co-factor value in 2nd Column of 1st Row . (we are evaluating co factors row wise and writing Column wise ) .
R₁ 5 -1 4 5 -1
R₂ 2 3 5 2 3
R₃ 5 -2 6 5 -2
R₄ 5 -1 4 5 -1
R5 2 3 5 2 3
Similarly for 1st , 2nd, 3rd element the co-factor values will be as follows
For A₃1 i.e 5
R₁ 5 -1 4 5 -1
R₂ 2 3 5 2 3
R₃ 5 -2 6 5 -2
R₄ 5 -1 4 5 -1
R5 2 3 5 2 3
For A₃₂ i.e -2
R₁ 5 -1 4 5 -1
R2 2 3 5 2 3
R₃ 5 -2 6 5 -2
R₄ 5 -1 4 5 -1
R5 2 3 5 2 3
for A₃₃ i.e. 6
R₁ 5 -1 4 5 -1
R₂ 2 3 5 2 3
R₃ 5 -2 6 5 -2
R₄ 5 -1 4 5 -1
R5 2 3 5 2 3
so we have -17 ,-17 and 17 as co-factors of 3rd Row, write these co factors in 3rd column .
(we are evaluating co factors row wise and writing Column wise)
Ad joint A =
⎾ 28 -2 -17 ⏋
⎹⎸ 13 10 -17 ⎹
⎿ -19 5 17 ⏌
Now divide with the determinant value of given 3×3 Matrix , which will be 5(28)-1(-13) + 4(-19) = 140 + 13 -7 6 = 77.
Now divide each element of Ad joint Matrix obtained in previous step with determinant value 77,
Then A⁻¹ =
How to find Inverse of 2×2 Matrix using elementary Transformations.Short cut to inverse of 2×2 .
Conclusion
This post was regarding short cut methods of finding Inverse of 2×2 and 3×3 Matrices , INVERSE OF 2×2 AND 3×3 MATRIX,
shortcut to find inverse of 2 × 2 matrix, transpose 3x3, inverse of determinant, matrices and determinants, determinant calculator, crammer's rule,If you liked this post ,Please share your precious views on this topic and share this post with your friends to benefit them. we shall Meet in the next post ,till then BYE .
