Let A be a 10 × 10 matrix such that A5 is a null matrix, and let I be the 10 × 10 identity matrix. The determinant of A + I is ______.
Concept:
Nilpotent Matrix: Any square matrix of order n is said to be nilpotent matrix if there exist least positive integer m such that Am = O, where O is the null matrix of order n.
The determinant of the sum of the nilpotent matrix with the identity matrix of the same order is always unity.
Example:
Consider a nilpotent matrix of order 2
\(A = \left[ {\begin{array}{*{20}{c}} 2&{-1}\\ { 4}&{ - 2} \end{array}} \right]\)
\({A^2} = \left[ {\begin{array}{*{20}{c}} 2&{-1}\\ { 4}&{ - 2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 2&{-1}\\ { 4}&{ - 2} \end{array}} \right]= \left[ {\begin{array}{*{20}{c}} 0&0\\ 0&0 \end{array}} \right] \)
So here A + I = \(\left[ {\begin{array}{*{20}{c}} 3&{-1}\\ {4}&{ -1} \end{array}} \right]\)
⇒ |A + I| = 1
Calculation:
Given A is a 10 × 10 matrix and A5 is a null matrix,
So, A is a nilpotent matrix of order 10.
Also given I is the 10 × 10 identity matrix.
Then the determinant of A + I = 1 (unity)
Additional Information
Singular Matrix: Any square matrix of order n is said to be singular if |A| = 0.
Involuntary Matrix: Any square matrix of order n is said to be an involuntary matrix if A2 = I, where I is the identity matrix of order n.
Idempotent Matrix: Any square matrix of order n is said to be an idempotent matrix if A2 = A.
Important Questions from Matrix Algebra
Consider the system of equations: x + y = 2 and 2x + 2y = 5. This system has
The standard ordered basis of R 3 is {e 1, e 2, e 3} Let T : R 3 → R 3 be the linear transformation such that T(e 1) = 7e 1- 5e 3, T (e 2) = -2e 2+ 9e 3, T(e 3) = e 1+ e 2+ e 3. The standard matrix of T is:
The system of equations
x + y + z = 6;
x + 4y + 6z = 20;
x + 4y + λz = μ
has NO solution for values of λ and μ given by
What is the transformation matrix M that transforms a square in the xy-plane defined by (1, 1) T, (-1, 1) T, (-1, -1) T and (1, -1) T to a parallelogram whose corresponding vertices are (2, 1) T, (0, 1) T, (-2, -1) T and (0, -1) T?
- If A = \( \left[\begin{array}{cc}0 & 1 \\ −1 & 0\end{array}\right]\) and (aI 2 + bA)2 = A, then
