Suppose matrix A- 2 T- -1 6 - and X is a 2 - 1 non-null column vector- then AX - -IX for what values of - I is the identity matrix of order 2

AX = λIX ⇒ (A λI)X = 0 Since X is not a null vector, |A λ I|=0 A λ I= [ 2 amp;7; 1 amp;6 ] [ λ amp;0; 0 amp; λ ] =[ 2 λ amp;7; ...

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Standard XI

Mathematics

Evaluation of a Determinant

Suppose matri...

Question

Suppose matrix $A = [\begin{matrix} 2 & T - 1 & 6 \end{matrix}]$ and X is a 2 $\times$ 1 non-null column vector, then AX = $λ$ IX for what value(s) of $λ$ ? (I is the identity matrix of order 2)

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Solution

The correct option is C
1

AX = $λ$ IX $\Rightarrow$ (A - $λ$ I)X = 0
Since X is not a null vector, $| A - λ I | = 0$
$A - λ I = [\begin{matrix} 2 & 7 - 1 & 6 \end{matrix}] - [\begin{matrix} λ & 0 0 & λ \end{matrix}] = [\begin{matrix} 2 - λ & 7 - 1 & 6 - λ \end{matrix}]$
$| A - λ I | = (2 - λ) (6 - λ) + 7 = λ^{2} + 4 λ - 5 = (λ + 5) (λ - 1) | A - λ I | = 0 \Rightarrow λ = - 5 o r λ = 1$
(b) and (c) are the correct options.

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Similar questions

Suppose matrix $A = [\begin{matrix} 2 & T - 1 & 6 \end{matrix}]$ and X is a 2 $\times$ 1 non-null column vector, then AX = $λ$ IX for what value(s) of $λ$ ? (I is the identity matrix of order 2)

Q. If

P

is a non null matrix of order

3

with real number as entries, such that

P^{3} = 0,

where

O

is null matrix of order

3

then (where

I

is the identity matrix of order

3

)

Let A be a square matrix of order 2 or 3 and I will be the identity matrix of the same order. Then the matrix A -

λ

I is called the characteristic matrix of the matrix A, where

λ

is some complex number. The determinant of the characteristic matrix is called characteristic determinant of matrix A which will, of course, be a polynomial of degree 3 in

λ

. The equation det (A -

λ

=

0 is called the characteristic equation of the matrix A and its roots (the values of

λ

) are called characteristic roots or eigenvalues. It is also known that every square matrix has its characteristic equation.

The eigenvalues of the matrix

A = ⎡ ⎢ ⎣ \begin{matrix} 2 & 1 & 1 2 & 3 & 4 - 1 & - 1 & - 2 \end{matrix} ⎤ ⎥ ⎦

are

Q. Let X be an unknown matrix of order 2 and I be an identity matrix of order 2, then the
equation

X^{2} = 1

Q. For two matrices A and B of same order, if AB = A and BA = B then

A^{2}

is equal to?

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Suppose matrix A=[2T−16] and X is a 2 × 1 non-null column vector, then AX = λIX for what value(s) of λ? (I is the identity matrix of order 2)

The correct option is C 1 AX = λIX ⇒ (A - λI)X = 0 Since X is not a null vector, |A−λI|=0 A−λI=[27−16]−[λ00λ]=[2−λ7−16−λ] |A−λI|=(2−λ)(6−λ)+7=λ2+4λ−5=(λ+5)(λ−1)|A−λI|=0 ⇒ λ=−5 or λ=1 (b) and (c) are the correct options.

Suppose matrix $A = [\begin{matrix} 2 & T - 1 & 6 \end{matrix}]$ and X is a 2 $\times$ 1 non-null column vector, then AX = $λ$ IX for what value(s) of $λ$ ? (I is the identity matrix of order 2)