Advertisements
Advertisements
Question
If A = `[(α, β),(γ, -α)]` is such that A2 = I, then ______.
Options
1 + α2 + βγ = 0
1 – α2 + βγ = 0
1 – α2 – βγ = 0
1 + α2 – βγ = 0
Advertisements
Solution
If A = `[(α, β),(γ, -α)]` is such that A2 = I, then 1 – α2 – βγ = 0.
Explanation:
A = `[(α, β), (γ, -α)]`
`A^2 = A * A [(α, β), (γ, -α)][(α, β), (γ, -α)]`
= `[(α^2 + βγ, αβ - αβ), (αγ - αγ, βγ + α^2)] = [(1, 0), (0, 1)]`
Now, A2 = I
⇒ `[(α^2 + βγ,0), (0, βγ + α^2)] = [(1, 0), (0, 1)]`
α2 + βγ = 1 or 1 – α2 – βγ = 0
Accordingly, option (1 – α2 – βγ = 0) is correct.
APPEARS IN
RELATED QUESTIONS
Let A = `[(0,1),(0,0)]`show that (aI+bA)n = anI + nan-1 bA , where I is the identity matrix of order 2 and n ∈ N
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 is a square matrix such that A2 = A, then (I + A)3 – 7A is equal to ______.
Let A = `((2,-1),(3,4))`, B = `((5,2),(7,4))`, C= `((2,5),(3,8))` find a matrix D such that CD − AB = O
Find the non-singular matrices P & Q such that PAQ is in normal form where`[(1,2,3,4),(2,1,4,3),(3,0,5,-10)]`
if `vec"a"= 2hat"i" + 3hat"j"+ hat"k", vec"b" = hat"i" -2hat"j" + hat"k" and vec"c" = -3hat"i" + hat"j" + 2hat"k", "find" [vec"a" vec"b" vec"c"]`
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:
`[(3, -2, 4),(0, 0, -5),(0, 0, 0)]`
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:
`[(3, 0, 0),(0, 5, 0),(0, 0, 1/3)]`
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:
`[(1, 0, 0),(0, 1, 0),(0, 0, 1)]`
Identify the following matrix is singular or non-singular?
`[(7, 5),(-4, 7)]`
Find k if the following matrix is singular:
`[(7, 3),(-2, "k")]`
If A = `[(5, 1, -1),(3, 2, 0)]`, Find (AT)T.
Answer the following question:
If A = `[(1, 2),(3, 2),(-1, 0)]` and B = `[(1, 3, 2),(4, -1, -3)]`, show that AB is singular.
State whether the following statement is True or False:
If A is non singular, then |A| = 0
If A = `[(3, 1),(-1, 2)]`, then prove that A2 – 5A + 7I = O, where I is unit matrix of order 2
Given A = `[(2, 4, 0),(3, 9, 6)]` and B = `[(1, 4),(2, 8),(1, 3)]` is (AB)′ = B′A′?
If `[("a","b"),("c", "-a")]`is a square root of the 2 x 2 identity matrix, then a, b, c satisfy the relation ____________.
The matrix `[(0,5,-7),(-5,0,11),(7,-11,0)]` is ____________.
A square matrix B = [bÿ] m × m is said to be a diagonal matrix if all diagonal elements are
How many matrices can be obtained by using one or more numbers from four given numbers?
If A is a square matrix of order 3, then |2A| is equal to ______.
Assertion: Let the matrices A = `((-3, 2),(-5, 4))` and B = `((4, -2),(5, -3))` be such that A100B = BA100
Reason: AB = BA implies AB = BA for all positive integers n.
What is the order of a column matrix with \[m\] rows?
A matrix with only one row is called a:
Which general form represents a row matrix?
What is the order of a row matrix with \[n\] columns?
What is the order of \[B=\begin{bmatrix}-\dfrac{1}{2}&\sqrt{5}&2&3\end{bmatrix}\]?
Which elements can be non-zero in a diagonal matrix?
A scalar matrix is a diagonal matrix in which:
Which statement correctly describes the relationship between scalar matrices and diagonal matrices?
How is an identity matrix denoted when its order is \[n\]?
What happens when any matrix is multiplied by an identity matrix?
A zero matrix (null matrix) is a matrix in which:
What is the notation for a zero matrix?
What role does the zero matrix play in matrix algebra?
