IPMAT Question Paper 2019 | IPM Indore Quants

IPMAT 2019 Question Paper IPM Indore Quantitative Ability. Solve questions from IPMAT 2019 Question Paper from IPM Indore and check the solutions to get adequate practice. The best way to ace IPMAT is by solving IPMAT Question Paper. To solve other IPMAT Sample papers, go here: IPM Sample Paper

Question 5 : If A is a 3 X 3 non-zero matrix such that A² = 0 then determinant of [(1 + A)² - 50A] is equal to

Explanatory Answer

A non-zero Square matrix A is said to be a nilpotent matrix of order n if Aⁿ = 0. Determinant of[(I+ A)⁵⁰ - 50A] should be the same for any nilpotent matrix of order 2.

So, Let us consider a simple 3 x 3 nilpotent matrix. A = \\left[\begin{array}{ccc}0 & 1 & 1\\0 & 0 & 0\\0 & 0 & 0\end{array}\right]\\)

Check that A² = 0.

I + A = \\left[\begin{array}{ccc}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{array}\right]\\) + \\left[\begin{array}{ccc}0 & 1 & 1\\0 & 0 & 0\\0 & 0 & 0\end{array}\right]\\) = \\left[\begin{array}{ccc}1 & 1 & 1\\0 & 1 & 0\\0 & 0 & 1\end{array}\right]\\).

Observe that I + A is an Upper triangular matrix with all the principal diagonal elements being 1.

(Upper triangular matrix, means that the elements below the diagonal, that is the elements on the South – East are all 0.)

Any number of times a triangular matrix is multiplied to itself, the result is always a triangular matrix.
Something more special about I + A can be known by keen observation, Since the principal diagonal elements are unity(1), and it is a upper triangular matrix, any number of times (I + A) is multiplied to itself, it will remain an Upper triangular matrix with all the principal diagonal elements maintained at unity(1).

Check (I + A)².

Therefore, (I + A)⁵⁰, will also be an Upper triangular matrix with all the principal diagonal elements being 1.

So, (I + A)⁵⁰ may look something like this

(I + A)⁵⁰ = \\left[\begin{array}{ccc}1 & x & y\\0 & 1 & z\\0 & 0 & 1\end{array}\right]\\)
50 A = 50 \\left[\begin{array}{ccc}0 & 1 & 1\\0 & 0 & 0\\0 & 0 & 0\end{array}\right]\\) = \\left[\begin{array}{ccc}0 & 50 & 50\\0 & 0 & 0\\0 & 0 & 0\end{array}\right]\\)
Observe that both (I + A)⁵⁰ and 50 A are Upper triangular matrices. So will their sum or difference be.
(I + A)⁵⁰ - 50 A = \\left[\begin{array}{ccc}1 & x & y\\0 & 1 & z\\0 & 0 & 1\end{array}\right]\\) - \\left[\begin{array}{ccc}0 & 50 & 50\\0 & 0 & 0\\0 & 0 & 0\end{array}\right]\\) = \\left[\begin{array}{ccc}1 & x-50 & y-50\\0 & 1 & z\\0 & 0 & 1\end{array}\right]\\)

Clearly, (I + A)⁵⁰ - 50 A is also an Upper triangular matrix with principal diagonal elements being 1.

We are asked to find the determinant of (I + A)⁵⁰ - 50 A ,
Determinant of any triangular matrix is the sum of its principal diagonal elements.
So, determinant of (I + A)⁵⁰ - 50 A = 1 + 1 + 1 = 3
Determinant of (I + A)⁵⁰ - 50 A = 3.

Tip:You should choose your nilpotent matrix very carefully, For this problem, It has to be a triangular matrix with Principal Diagonal Elements being 0 too.

The question is "If A is a 3 X 3 non-zero matrix such that A² = 0 then determinant of [(1 + A)² - 50A] is equal to"

Hence, the answer is 3

Verbal Ability for CAT | CAT Reading Comprehension

Verbal Ability for CAT| CAT Para Jumble

Verbal Ability for CAT | CAT Sentence Elimination

Verbal Ability for CAT | CAT Paragraph Completion

Verbal Ability for CAT | CAT Critical Reasoning

Verbal Ability for CAT | CAT Word Usage

Verbal Ability for CAT| CAT Para Summary

Verbal Ability for CAT | CAT Text Completion

Verbal Ability for CAT | CAT Sentence Correction