Advertisements
Advertisements
Question
If A is a square matrix such that A2 = I, then find the simplified value of (A – I)3 + (A + I)3 – 7A.
Advertisements
Solution
Given:
(A−I)3+(A+I)3−7A
=A3−I3−3A2I+3AI2+A3+I3+3A2I+3AI2−7A
= 2A3+6AI2−7A
=2A.A2+6AI2−7A
=8A−7A
=A
APPEARS IN
RELATED QUESTIONS
Find the value of x, y, and z from the following equation:
`[(x + y + z), (x + z), (y + z)] = [(9), (5), (7)]`
Find the value of a, b, c and d from the equation:
`[(a - b, 2a + c),(2a - b, 3c + d)] = [(-1, 5),(0, 13)]`
If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N
If A = `[(α, β),(γ, -α)]` is such that A2 = I, then ______.
Given `A = [(2,-3),(-4,7)]` compute `A^(-1)` and show that `2A^(-1) = 9I - A`
In a certain city there are 30 colleges. Each college has 15 peons, 6 clerks, 1 typist and 1 section officer. Express the given information as a column matrix. Using scalar multiplication, find the total number of posts of each kind in all the colleges.
If li, mi, ni, i = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AAT = I, where \[A = \begin{bmatrix}l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3\end{bmatrix}\]
If\[A = \begin{bmatrix}2 & 3 \\ 4 & 5\end{bmatrix}\]prove that A − AT is a skew-symmetric matrix.
If A and B are square matrices of the same order 3, such that ∣A∣ = 2 and AB = 2I, write the value of ∣B∣.
Classify the following matrix as, a row, a column, a square, a diagonal, a scalar, a unit, an upper triangular, a lower triangular, a symmetric or a skew-symmetric matrix:
`[(5),(4),(-3)]`
Identify the following matrix is singular or non-singular?
`[(3, 5, 7),(-2, 1, 4),(3, 2, 5)]`
Identify the following matrix is singular or non-singular?
`[(7, 5),(-4, 7)]`
The following matrix, using its transpose state whether it is symmetric, skew-symmetric, or neither:
`[(2, 5, 1),(-5, 4, 6),(-1, -6, 3)]`
Choose the correct alternative:
If B = `[(6, 3),(-2, "k")]` is singular matrix, then the value of k is ______
If A = `[(3, 1),(-1, 2)]`, then prove that A2 – 5A + 7I = O, where I is unit matrix of order 2
If two matrices A and B are of the same order, then 2A + B = B + 2A.
For the non singular matrix A, (A′)–1 = (A–1)′.
If A = `[(3, -4),(1, 1),(2, 0)]` and B = `[(2, 1, 2),(1, 2, 4)]`, then verify (BA)2 ≠ B2A2
Show by an example that for A ≠ O, B ≠ O, AB = O
If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.
2X + 3Y = `[(2, 3),(4, 0)]`, 3Y + 2Y = `[(-2, 2),(1, -5)]`
If A = `[(0,0,0),(0,0,0),(0,1,0)]` then A is ____________.
If A `= [("cos x", - "sin x"),("sin x", "cos x")]`, find AAT.
`[(5sqrt(7) + sqrt(7)) + (4sqrt(7) + 8sqrt(7))] - (19)^2` = ?
A matrix is said to be a column matrix if it has
If the sides a, b, c of ΔABC satisfy the equation 4x3 – 24x2 + 47x – 30 = 0 and `|(a^2, (s - a)^2, (s - a)^2),((s - b)^2, b^2, (s - b)^2),((s - c)^2, (s - c)^2, c^2)| = p^2/q` where p and q are co-prime and s is semiperimeter of ΔABC, then the value of (p – q) is ______.
How many matrices can be obtained by using one or more numbers from four given numbers?
