A matrix is a specific collection of elements organised in rows and columns. A matrix’s order is the number of rows x the number of columns. For instance, a 2*2 matrix means it contains two rows and two columns. We can only discover the matrix inverse for square matrices with equal numbers of rows and columns. The Inverse of a matrix is used to solve linear equations using the matrix inversion method.
The inverse of a matrix X is represented by X-1. A simple formula may be used to determine the inverse of a 2*2 matrix. The inverse of a matrix is another matrix that yields the multiplicative identity when multiplied with the supplied matrix.
Let us now look at the formula, methods, and terminologies associated with the inverse of a matrix.
What is the Inverse of the Matrix?
The Inverse of a matrix is another matrix that yields the multiplicative identity when multiplied by the supplied matrix. X-1 is the inverse of a matrix X, where X.X-1 = X-1. X = I, where I is the identity matrix. An invertible matrix has a non-zero determinant and for which the inverse matrix may be determined.
For example, the inverse of X =
Is
Properties of Inverse Matrix
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The original matrix is equal to the inverse of the inverse matrix.
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If A and B are invertible matrices, then AB must be as well. As a result, (AB)-1 = B-1A-1
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If A is nonsingular, then (AT)-1 = (A-1)T
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A matrix’s product with its inverse, and vice versa, is always equal to the identity matrix.
Inverse Matrix Formula
In the case of real numbers, the inverse of every real number a was the number a-1, so that a multiplied by a-1 equaled 1. We understood that the inverse of a real number was the reciprocal of the number, as long as the number was not zero. The matrix is the inverse of a square matrix X, denoted by X-1, such that the product of X and X-1 equals the identity matrix. The resulting identity matrix will be the same size as matrix X.
A Matrix’s inverse:
X-1 = 1/|X|
Because |X| is in the denominator of the expression, the inverse of the matrix exists only if the determinant of the matrix is non-zero. That is to say, |X|= 0.
The Inverse of Matrix Question
Some students may find it difficult to answer the Inverse of 3 by 3 Matrix. Hence, our question of the inverse of a matrix will be of a 3 by 3 matrix.
Steps to follow to find the inverse of a 3 by 3 matrix:
Calculate the determinant of a given matrix.
The first step is to compute the determinant of the 3 * 3 matrix, then discover its cofactors, minors, and adjoint, and then incorporate the findings into the inverse matrix formula shown below.
X−1=1/|X|Adj(X)
For example:
Check to see if the following matrix is invertible.
This may be demonstrated if the determinant is non-zero. There will be no inverse of the provided matrix if the determinant of the supplied matrix is zero.
det(X) = 1(0-24) – 2(0-20) + 3(0-5)
det(X) = -24 + 40 – 15
det (X) = 1
As the value determinant is 1, we can say that a given matrix has an inverse matrix.
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Determine the matrix’s transpose.
To determine the transposition of the given 3 by 3 matrix.
Hence, XT =
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Determine the determinant of the two-by-two matrix.
We will now calculate the determinant of each 2 X 2 minor matrices.
For 1st row elements:
Equals to -24
Equals to -18
Equals to 5
For 2nd row elements:
Equals to -20
Equals to -15
Equals to 4
For 3rd row elements:
Equals to -5
Equals to -4
Equals to 1
The new matrix is:
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Create the cofactor matrix.
Reverse the sign of the alternating terms to get the adjoint or adjugate matrix, as shown below:
As a result, we have the new matrix, X:
Adj (X) = the new matrix is :
Multiplied by :
Adj (X) =
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Finally, divide each adjugate matrix term by the determinant.
Determining the inverse of a 3 x 3 matrix:
Now, in the formula, we may swap the values of det (X) and adj (X):
X−1 = (1/det (X)) Adj (X)
The inverse of the matrix is: X−1 = (1/1) =
Conclusion
By now you would have understood how to inverse a matrix and what are the steps involved in it. Typically you need to first find the determinant of a matrix. Then calculate the cofactor matrix. And, finally divide each adjugate matrix term by the determinant. The inverse of a Matrix is an important topic from the point of view of several competitive exams. Make sure to have a good grasp of this topic.