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What we mean by nilpotent matrix?
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What we mean by nilpotent matrix?
When A^k = 0 while k is the index of nilpotent matrix
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What we mean by nilpotent matrix?
When A^2 = 0
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Introduction:To find the inverse of a matrix A using elementary transformations, we need to perform a series of row operations until A is transformed into the identity matrix I. Simultaneously, we perform the same row operations on the identity matrix I and obtain the inverse matrix A^-1.Given Matrix:A = [1 2 2 -1]Augmented Matrix:We will augment the given matrix A with the identity matrix I as follows:[A | I] = [1 2 2 -1 | 1 0 0 1]Row Operations:Perform the following row operations to transform A into I:1. R2 = R2 - 2R1[A | I] = [1 2 2 -1 | 1 0 0 1] [0 -2 -2 1 | -2 0 0 0] 2. R2 = -1/2R2[A | I] = [1 2 2 -1 | 1 0 0 1] [0 1 1/2 -1/2 | 1 0 0 0] 3. R1 = R1 - 2R2[A | I] = [1 0 1 -2 | -1 0 0 1] [0 1 1/2 -1/2 | 1 0 0 0] 4. R1 = R1 + R2[A | I] = [1 0 3/2 -5/2 | 0 0 0 1] [0 1 1/2 -1/2 | 1 0 0 0] Final Result:After performing the row operations, the matrix A is transformed into the identity matrix I. The inverse matrix A^-1 is given by the augmented matrix on the right side:A^-1 = [0 0 0 1 | 0 0 0 1] [0 1 1/2 -1/2 | 1 0 0 0]Explanation:By using elementary transformations, we performed a series of row operations on the given matrix A to transform it into the identity matrix I. Simultaneously, we performed the same row operations on the identity matrix I to obtain the inverse matrix A^-1. These row operations include adding or subtracting multiples of one row from another and multiplying a row by a constant. These operations ensure that the resulting matrix A^-1, when multiplied with the original matrix A, yields the identity matrix I. Therefore, A^-1 is the inverse of matrix A.

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What we mean by nilpotent matrix?
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