Show that 2 − 2 − 4 − 1 3 4 1 − 2 − 3 is idempotent.

Let A = 2 − 2 − 4 − 1 3 4 1 − 2 − 3 A matrix A is said to idempotent if A 2 = A Now, A 2 = A A = 2 - 2 - 4 - 1 3 4 1 - 2 - 3 2 − 2 − 4 − 1 3 4 1 − 2 − 3 = 4 + 2 - 4 - 4 - 6 + 8 - 8 - 8 + 12 - 2 - 3 + 4 2 + 9 - 8 4 + 12 - 12 2 + 2 - 3 - 2 - 6 + 6 - 4 - 8 + 9 = 2 − 2 − 4 − 1 3 4 1 − 2 − 3 = A Hence, A 2 = A .

If A = 0 1 − 1 4 − 3 4 3 − 3 4 , then show that A 2 = I 3 .

Given, A = 0 1 - 1 4 - 3 4 3 - 3 4 Now, A 2 = 0 1 - 1 4 - 3 4 3 - 3 4 0 1 - 1 4 - 3 4 3 - 3 4 = 0 × 0 + 1 × 4 + - 1 3 0 × 1 + 1 - 3 + - 1 - 3 0 - 1 + 1 × 4 + - 1 4 4 × 0 + - 3 4 + 4 × 3 4 × 1 + - 3 - 3 + 4 - 3 4 - 1 + - 3 4 + 4 × 4 3 × 0 + - 3 4 + 4 × 3 3 × 1 + - 3 - 3 + 4 - 3 3 - 1 + - 3 4 + 4 × 4 = 0 + 4 - 3 0 - 3 + 3 0 + 4 - 4 0 - 12 + 12 4 + 9 - 12 - 4 - 12 + 16 0 - 12 + 12 3 + 9 - 12 - 3 - 12 + 16 = 1 0 0 0 1 0 0 0 1 = I 3 Hence, proved.

If A = 3 − 4 1 − 1 and n be a positive integer, then prove that A n = 1 + 2 n − 4 n n 1 − 2 n .

Given, A = 3 - 4 1 - 1 Now, A 2 = 3 - 4 1 - 1 3 - 4 1 - 1 = 9 - 4 - 12 + 4 3 - 1 - 4 + 1 = 5 - 8 2 - 3 = 1 + 2 2 - 4 2 2 1 - 2 2 A 3 = A · A 2 = 3 - 4 1 - 1 5 - 8 2 - 3 = 15 - 8 - 24 + 12 5 - 2 - 8 + 3 = 7 - 12 3 - 5 = 1 + 2 3 - 4 3 3 1 - 2 3 Therefore, A n = 1 + 2 n - 4 n n 1 - 2 n Hence, proved.

Show that λ 1 0 λ n = λ n n λ n − 1 0 λ n ( λ ≠ 0 ) for any + v e integer n . Hence obtain the value of A 51 , where A = 1 1 0 1 .

Given, λ 1 0 λ n = λ n n λ n − 1 0 λ n For, n = 1 λ 1 0 λ 1 = λ 1 1 λ 1 − 1 0 λ 1 = λ 1 0 λ For, n = 2 λ 1 0 λ 2 = λ 1 0 λ λ 1 0 λ = λ 2 + 0 λ + λ 0 + 0 0 + λ 2 = λ 2 2 λ 0 λ 2 = λ 2 2 λ 2 - 1 0 λ 2 For n = 3 λ 1 0 λ 3 = λ 1 0 λ 2 λ 1 0 λ = λ 2 2 λ 0 λ 2 λ 1 0 λ = λ 3 + 0 λ 2 + 2 λ 2 0 + 0 0 + λ 3 = λ 3 2 λ 2 0 λ 3 = λ 3 2 λ 3 - 1 0 λ 3 ∴ λ 1 0 λ n = λ n n λ n − 1 0 λ n ( λ ≠ 0 ) Hence, proved. Now, put λ = 1 The resultant matrix is 1 1 0 1 = A Now, A 51 = 1 1 0 1 51 = 1 51 51 1 51 - 1 0 1 51 ⇒ A 51 = 1 51 0 1 .

E M B I B E

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 3 - Elementary Theory of Matrices>EXERCISE>Q 24

EASY

12th West Bengal Board

IMPORTANT

Earn 100

If $A$ and $B$ are two square matrices such that $A B = B$ and $B A = A$ ; then show that $A$ and $B$ both are idempotent.

Important Questions on Elementary Theory of Matrices

MEDIUM

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 3 - Elementary Theory of Matrices>EXERCISE>Q 24

Show that

(\begin{array}{ccc} 2 & - 2 & - 4 \\ - 1 & 3 & 4 \\ 1 & - 2 & - 3 \end{array})

is idempotent.

HARD

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 3 - Elementary Theory of Matrices>EXERCISE>Q 25

Express each of the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix

$(\begin{matrix} 3 & 2 & 3 \\ 4 & 5 & 3 \\ 2 & 4 & 5 \end{matrix})$

HARD

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 3 - Elementary Theory of Matrices>EXERCISE>Q 25

Express each of the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix

$(\begin{array}{lll} 3 2 5 \\ 4 1 3 \\ 0 6 7 \end{array})$

HARD

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 3 - Elementary Theory of Matrices>EXERCISE>Q 25

Express each of the following matrices as the sum of a symmetric matrix and a skew-symmetric matrix

$(\begin{array}{ccc} - 3 & 4 & 1 \\ 2 & 3 & 0 \\ 1 & 4 & 5 \end{array})$

MEDIUM

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 3 - Elementary Theory of Matrices>EXERCISE>Q 25

Express the following matrix as the sum of a symmetric and a skew-symmetric matrix $[\begin{array}{r} 1 & 3 & 5 \\ - 6 & 8 & 3 \\ - 4 & 6 & 5 \end{array}]$