Have a query? Ask a Tutor!!

Access to best educators and exclusive paper discussions

Can’t find what you’re looking for?

Our tutors are from top schools around the world, and they are waiting for you 24/7, anytime, anywhere.

Homework- One to One Solutions
Understand concepts to solve better
Get your doubts solved instantly

Maths-

General

Easy

Question

If A^k = 0, for some value of k, (I – A)^p = I + A + A² + …. A^k–1, thus p is (A is nilpotent with index k).

–1
–2
–3
None of these

The correct answer is: –1

Let B = I + A + A² + … + A^k–1
Post multiply both sides by (I – A), so that
B(I – A) = (I + A + A² + … + A^k–1) (I – A)
= I – A + A – A² + A² – A³ + … –A^k–1 + A^k–1 – A^k
= I – A^k = I, since A^k = 0
$rightwards double arrow$ B = (I – A)^–1
Hence (I – A)^–1 = I + A + A² + … + A^k–1.
Thus, p = –1.
Hence (A) is correct answer.

Related Questions to study

If the matrix A = $open square brackets table row cell y plus a end cell b c row a cell y plus b end cell c row a b cell y plus c end cell end table close square brackets$ has rank 3, then -

If the matrix A = $open square brackets table row cell y plus a end cell b c row a cell y plus b end cell c row a b cell y plus c end cell end table close square brackets$ has rank 3, then -

If A is non-singular matrix of order 3 × 3, then adj (adj A) is equal to -

If A is non-singular matrix of order 3 × 3, then adj (adj A) is equal to -

For k = $blank fraction numerator 1 over denominator square root of 50 end fraction$ , the value of a, b, c such that PP' = I, where P = $open square brackets table row cell 2 divided by 3 end cell cell 3 k end cell a row cell negative 1 divided by 3 end cell cell negative 4 k end cell b row cell 2 divided by 3 end cell cell negative 5 k end cell c end table close square brackets$ is-

For k = $blank fraction numerator 1 over denominator square root of 50 end fraction$ , the value of a, b, c such that PP' = I, where P = $open square brackets table row cell 2 divided by 3 end cell cell 3 k end cell a row cell negative 1 divided by 3 end cell cell negative 4 k end cell b row cell 2 divided by 3 end cell cell negative 5 k end cell c end table close square brackets$ is-

parallel

If matrix A = $open square brackets table row a b c row b c a row c a b end table close square brackets$ where a, b, c are real positive numbers, abc = 1 and A^T A = 1, then the value of a³ + b³ + c³ is -

If matrix A = $open square brackets table row a b c row b c a row c a b end table close square brackets$ where a, b, c are real positive numbers, abc = 1 and A^T A = 1, then the value of a³ + b³ + c³ is -

If [x] stands for the greatest integer less or equal to x, then in order than the set of equations x – 3y = 4 ; 5x + y = 2 ; [2 $pi$ ]x – [e]y = [2a] may be consistent, then ‘a’ should lie in -

If [x] stands for the greatest integer less or equal to x, then in order than the set of equations x – 3y = 4 ; 5x + y = 2 ; [2 $pi$ ]x – [e]y = [2a] may be consistent, then ‘a’ should lie in -

If the matrix $open square brackets table row 0 cell 2 beta end cell gamma row alpha beta cell negative gamma end cell row alpha cell negative beta end cell gamma end table close square brackets$ is orthogonal, then -

If the matrix $open square brackets table row 0 cell 2 beta end cell gamma row alpha beta cell negative gamma end cell row alpha cell negative beta end cell gamma end table close square brackets$ is orthogonal, then -

parallel

For each real x such that $negative 1 less than x less than 1$ . Let A (x) denote the matrix (1 – x)^–1/2 $open parentheses table row 1 cell negative x end cell row cell negative x end cell 1 end table close parentheses$ and $A open parentheses x close parentheses A open parentheses y close parentheses equals k A open parentheses z close parentheses blank$ where $x comma y element of R comma negative 1 less than x comma y less than 1$ and z = $fraction numerator x plus y over denominator 1 plus x y end fraction$ then k is -

For each real x such that $negative 1 less than x less than 1$ . Let A (x) denote the matrix (1 – x)^–1/2 $open parentheses table row 1 cell negative x end cell row cell negative x end cell 1 end table close parentheses$ and $A open parentheses x close parentheses A open parentheses y close parentheses equals k A open parentheses z close parentheses blank$ where $x comma y element of R comma negative 1 less than x comma y less than 1$ and z = $fraction numerator x plus y over denominator 1 plus x y end fraction$ then k is -

If the system of equations x + 2y – 3z = 2, (k + 3)z = 3, (2k + 1)y + z = 2 is inconsistent, then value of k can be -

If the system of equations x + 2y – 3z = 2, (k + 3)z = 3, (2k + 1)y + z = 2 is inconsistent, then value of k can be -

If A is a 3 × 3 skew-symmetric matrix, then |A| is given by -

If A is a 3 × 3 skew-symmetric matrix, then |A| is given by -

parallel

The solubility of iodine in water is greatly increased by:

The solubility of iodine in water is greatly increased by:

chemistry-General

Let A = $open square brackets table row 2 3 row cell negative 1 end cell 5 end table close square brackets$ if A^–1 = xA + yI₂ then x and y are respectively -

Let A = $open square brackets table row 2 3 row cell negative 1 end cell 5 end table close square brackets$ if A^–1 = xA + yI₂ then x and y are respectively -

The value of a for which the system of equationsx + y + z = 0ax + (a + 1)y + (a + 2)z = 0 a³x + (a + 1)³y + (a + 2)³z = 0has a non-zero solution is -

The value of a for which the system of equationsx + y + z = 0ax + (a + 1)y + (a + 2)z = 0 a³x + (a + 1)³y + (a + 2)³z = 0has a non-zero solution is -

parallel

If A is non-singular matrix of order 3 × 3, then adj (adj A) is equal to -

If A is non-singular matrix of order 3 × 3, then adj (adj A) is equal to -

If A = $open square brackets table row a b c row x y z row p q r end table close square brackets$ , B = $open square brackets table row q cell negative b end cell y row cell negative p end cell a cell negative x end cell row r cell negative c end cell z end table close square brackets$ then -

If A = $open square brackets table row a b c row x y z row p q r end table close square brackets$ , B = $open square brackets table row q cell negative b end cell y row cell negative p end cell a cell negative x end cell row r cell negative c end cell z end table close square brackets$ then -

Let A, B, C be three square matrices of the same order, such that whenever AB = AC then B = C, if A is -

Let A, B, C be three square matrices of the same order, such that whenever AB = AC then B = C, if A is -

parallel

card img

With Turito Academy.

card img

With Turito Foundation.

card img

Get an Expert Advice From Turito.

Turito Academy

card img

With Turito Academy.

Test Prep

card img

With Turito Foundation.