Let A=( 1 2 0 0 1 3 1 -1 -8 ). Find the right inverse of

Let A=( 1 2 0 0 1 3 1 -1...

Key Concepts

Solving Linear Systems for Matrix Inversion

One effective method to compute the inverse of a matrix is by solving a set of linear systems derived from the equation A • X = I. By treating each column of the identity matrix as a separate right-hand side, one can solve for the corresponding columns of X. Techniques such as Gaussian elimination or other forms of systematic row operations are often used to obtain these solutions efficiently.

Verification of Inverse

After computing a candidate inverse, it is crucial to verify that the computed matrix functions correctly as both a left and a right inverse. This is done by checking that both A • X and X • A equal the identity matrix. Successful verification confirms that all the properties of the inverse are satisfied and that the solution is correct.

Matrix Inverse

A matrix inverse is a fundamental concept in linear algebra. For a square matrix A, its inverse (denoted A?¹) is another matrix such that when multiplied together in either order (A • A?¹ or A?¹ • A), the result is the identity matrix. The existence of an inverse is equivalent to the condition that A is non-singular, meaning its determinant is non-zero.

Right Inverse and Left Inverse

The concepts of right and left inverses refer to matrices that, when multiplied with the original matrix from the right or left side, yield the identity matrix. Specifically, if X is a right inverse of A, then A • X = I, and if X is a left inverse, then X • A = I. For square, non-singular matrices, the right inverse and left inverse coincide, meaning that if one exists the other does as well, and the matrix is completely invertible.

Transcript

00:01 We have given matrix a equal to matrix 1, 2 minus 2, minus 1 3 0 minus 2 1.

00:11 We need to verify that determinant of a inverse equal to 1 by determinant of a.

00:18 For this we need to first find a inverse, then find the determinant of a inverse, then determinant of a inverse, then determinant of a.

00:25 First we can find two determinant of a that is equal to 1 into 3 minus 0 then minus 2 into minus 1 minus 0 minus 2 into okay that is 3 minus 2 into minus 1 that is plus 2 minus 4 5 minus 4 equal to 1 okay determine 8 equal to 1 now we need to find a inverse.

01:09 A inverse.

01:10 We note the formula inverse equal to adjoined of a by determinant of a.

01:17 And where adjoined of a is the transpose of co -factor matrix.

01:24 Co -factor matrix.

01:27 Find co -factor matrix.

01:28 Then take the transpose.

01:30 We get adjoined of a.

01:32 So co -factor is the sine to minus.

01:35 Okay so a 1 1 equal to minus 1 raise 2 1 plus 1 this 1 plus 1 and minor of a 1 1 that is 3 so a 1 1 1 is 3 this is the co -factor of element a 1 1 okay likewise a 1 2 is minus 1 raise 2 1 plus 2 into and the minor of a 1 2 yes minus 1 so we get 1 minus here minus minus minus minus plus then a 1 3 equal to minus 1 raise to 4 into 2 that is 2 a 2 1 is minus 1 raise to 2 plus 1 into minus 2 that is 2 a 2 2 2 is 2 2 2 is minus 2 is minus 2 is minus 2 2 is minus 1 raised to 4 into 1 that is again 1 a2, 3 is minus 1 raised to 5 into minus 2.

02:45 So minus minus 2, that is 2.

02:48 Now a31 is minus 1 raised to 4 into 6.

03:01 That is 6 itself...

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