Advertisements
Advertisements
Question
If A is square matrix such that A2 = A, show that (I + A)3 = 7A + I..
Solution
We know that,
A . I = I . A
So, A and I are commutative.
Thus, we can expand (I + A)3 like real numbers expansion.
So, (I + A)3 = I3 + 3I2A + 3IA2 + A3
= I + 3IA + 3A2 + AA2 .....(As In = I, n ∈ N)
= I + 3A + 3A + AA
= I + 3A + 3A + A2
= I + 3A + 3A + A
= I + 7A
APPEARS IN
RELATED QUESTIONS
If \[\begin{pmatrix}a + 4 & 3b \\ 8 & - 6\end{pmatrix} = \begin{pmatrix}2a + 2 & b + 2 \\ 8 & a - 8b\end{pmatrix},\] ,write the value of a − 2b.
If A is a square matrix such that A2 = I, then (A – I)3 + (A + I)3 –7A is equal to ______.
Matrix addition is associative as well as commutative.
Matrix multiplication is commutative.
If A and B are two square matrices of the same order, then A + B = B + A.
(AB)–1 = A–1. B–1, where A and B are invertible matrices satisfying commutative property with respect to multiplication.
If `"A" = [("a","b"),("b","a")]` and `"A"^2 = [(alpha,beta),(beta, alpha)]` then ____________.
If matrix A `= [("a","b","c"),("b","c","a"),("c","a","b")]` where a, b, c are real positive numbers, abc = 1 and ATA = I, then the value of a3 + b3 + c3 is ____________.
Find the values of x, y, z respectively if the matrix A `= [(0,2"y","z"),("x","y","-z"),("x","-y","z")]` satisfy the equation ATA = I3.
If matrices A and B are inverse of each other then ____________.
If A `= [(5, "x"),("y", 0)]` and A = A' then ____________.
Matrices A and B will be inverse of each other only if
