how to find the inverse of matrix
Answers
Answer:
Conclusion
1.The inverse of A is A-1 only when A × A-1 = A-1 × A = I.
2.To find the inverse of a 2x2 matrix: swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc).
3.Sometimes there is no inverse at all.
Step-by-step explanation:
How to Find the Inverse of 3 x 3 Matrix?
1.Compute the determinant of the given matrix.
2.Take the transpose of the given matrix.
Calculate the determinant of 2×2 minor matrices.
3.Formulate the matrix of cofactors.
4.Finally, divide each term of the adjugate matrix by the determinant.
Answer:
How to Find the Inverse of 3 x 3 Matrix?
The steps to find the inverse of 3 by 3 matrix
Compute the determinant of the given matrix
Take the transpose of the given matrix
Calculate the determinant of 2×2 minor matrices
Formulate the matrix of cofactors
Finally, divide each term of the adjugate matrix by the determinant
Inverse Matrix Formula
First, find the determinant of 3 × 3Matrix and then find it’s minor, cofactors and adjoint and insert the results in the Inverse Matrix formula given below:
A−1=1|A|Adj(A)
Where |A| ≠ 0
Inverse of a 3 x 3 Matrix Example
Let’s see how 3 x 3 matrix looks :
M = ⎡⎣⎢adgbehcfi⎤⎦⎥
Consider the given 3×3 matrix:
A=⎡⎣⎢105216340⎤⎦⎥
Let’s see what are the steps to find Inverse.
Check the Given Matrix is Invertible
This can be proved if its determinant is non zero. If the determinant of the given matrix is zero, then there is no inverse for the given matrix
det (A) = 1(0-24) -2(0-20) + 3(0-5)
det(A) = -24 +40-15
det (A) = 1
Thus, we can say that the given matrix has an inverse matrix.
Taking the Transpose of the Given Matrix
Now take the transpose of the given 3×3 matrix.
Thus, A−1=⎡⎣⎢123014560⎤⎦⎥
Finding the Determinants of the 2×2 Minor Matrices
Now, we have to find the determinants of each and every 2×2 minor matrices
For first row elements:
[1460]=−24
[2360]=−18
[2314]=5
For second row elements:
[0450]=−20
[1350]=−15
[1304]=4
For third row elements:
[0156]=−5
[1256]=−4
[1201]=1
Now, the new matrix formed is:
A=⎡⎣⎢−24−20−5−18−15−4541⎤⎦⎥
Formulating the Matrix of Cofactors
Now, to create the adjoint or the adjugated matrix, reverse the sign of the alternating terms as shown below:
The obtained matrix is A=⎡⎣⎢−24−20−5−18−15−4541⎤⎦⎥
Adj (A) = ⎡⎣⎢−24−20−5−18−15−4541⎤⎦⎥×⎡⎣⎢+−+−+−+−+⎤⎦⎥
Adj (A) =⎡⎣⎢−2420−518−1545−41⎤⎦⎥
Finding the Inverse of the 3×3 Matrix
Now, substitute the value of det (A) and the adj (A) in the formula:
A-1 = [1/det(A)]Adj(A)
A-1 = (1/1)⎡⎣⎢−2420−518−1545−41⎤⎦⎥
Thus, the inverse of the given matrix is:
A-1 = (1/1)⎡⎣⎢−2420−518−1545−41⎤⎦⎥