Advertisements
Advertisements
प्रश्न
If A is square matrix such that A2 = A, show that (I + A)3 = 7A + I..
Advertisements
उत्तर
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
संबंधित प्रश्न
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.
Matrices A and B will be inverse of each other only if
Let \[A=[a_{ij}]\] and \[B=[b_{ij}]\] be two matrices of the same order \[m\times n\]. If \[C=A+B=[c_{ij}]\], which entry rule defines \[C\]?
For matrices \[A=[a_{ij}]\] and \[B=[b_{ij}]\] of the same order \[m\times n\], what is the order of \[C=A+B\]?
Let \[A=[a_{ij}]\] and \[B=[b_{ij}]\] be matrices of the same order \[m\times n\]. If \[D=A-B=[d_{ij}]\], which rule defines \[D\]?
Which expression is equivalent to \[A-B\]?
When can matrices be added or subtracted?
Which equation states the associative property of matrix addition?
Which equation expresses the additive identity property for a matrix \[A\]?
Which equation expresses the additive inverse property for a matrix \[A\]?
Given \[A=\begin{bmatrix}\sqrt{3}&1&-1\\2&3&0\end{bmatrix}\] and \[B=\begin{bmatrix}2&\sqrt{5}&1\\-2&3&\frac{1}{2}\end{bmatrix}\], find \[A+B\].
For \[A=\begin{bmatrix}1&2&3\\2&3&1\end{bmatrix}\] and \[B=\begin{bmatrix}3&-1&3\\-1&0&2\end{bmatrix}\], which matrix is \[-B\]?
If \[A=\begin{bmatrix}1&2&3\\2&3&1\end{bmatrix}\] and \[B=\begin{bmatrix}3&-1&3\\-1&0&2\end{bmatrix}\], find \[2A-B\].
