Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

JEE: Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

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Matrices


Miscellaneous Solutions


Question 1:
Let Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE where I is the identity
matrix of order 2 and n ε N


Answer
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have:

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE


Question 2:
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE


Answer

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

 


Question 3:
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
 

Answer
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

 


Question 4:
If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix.


Answer

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
Thus, (AB − BA) is a skew-symmetric matrix.

 


Question 5:
Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric
or skew symmetric.


Answer
We suppose that A is a symmetric matrix, then A' = A… (1)
Consider

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
Thus, if A is a skew-symmetric matrix, then B'AB is a skew-symmetric matrix.
Hence, if A is a symmetric or skew-symmetric matrix, then B'AB is a symmetric or
skewsymmetric matrix accordingly.

 


Question 6:
Solve system of linear equations, using matrix method.

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE


Answer

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE


Question 7:
For what values of Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE


Answer
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE


Question 8:
If

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE


Answer
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

 


Question 9:
Find x, if Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE


Answer
We have:

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

 


Question 10:
A manufacturer produces three products x, y, z which he sells in two markets.
Annual sales are indicated below:
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
(a) If unit sale prices of x, y and z are Rs 2.50, Rs 1.50 and Rs 1.00, respectively,
find the total revenue in each market with the help of matrix algebra.
(b) If the unit costs of the above three commodities are Rs 2.00, Rs 1.00 and 50
paise respectively. Find the gross profit.


Answer
(a) The unit sale prices of x, y, and z are respectively given as Rs 2.50, Rs 1.50, and Rs 1.00.
Consequently, the total revenue in market I can be represented in the form of a matrix
as:

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
The total revenue in market II can be represented in the form of a matrix as:
Therefore, the total revenue in market I isRs 46000 and the same in market II isRs
53000.
(b) The unit cost prices of x, y, and z are respectively given as Rs 2.00, Rs 1.00, and 50
paise.
Consequently, the total cost prices of all the products in market I can be represented in
the form of a matrix as:

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
Since the total revenue in market I isRs 46000, the gross profit in this marketis (Rs
46000 − Rs 31000) Rs 15000.
The total cost prices of all the products in market II can be represented in the form of a
matrix as:

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
Since the total revenue in market II isRs 53000, the gross profit in this market is (Rs
53000 − Rs 36000) Rs 17000.


Question 11:
Find the matrix X so that
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE


Answer
It is given that:

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
The matrix given on the R.H.S. of the equation is a 2 × 3 matrix and the one given on
the L.H.S. of the equation is a 2 × 3 matrix. Therefore, X has to be a 2 × 2 matrix.
Now, let Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
Therefore, we have:

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
Equating the corresponding elements of the two matrices, we have: Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

 


Question 12:
If A and B are square matrices of the same order such that AB = BA, then prove by
induction that  
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE for all n ε N


Answer
A and B are square matrices of the same order such that AB = BA.

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
Therefore, the result is true for n = k + 1.
Thus, by the principle of mathematical induction, we have Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE , for all natural
numbers.


Question 13:
Choose the correct answer in the following questions:

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE


Answer: C

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE


Question 14:
If the matrix A is both symmetric and skew symmetric, then
A. A is a diagonal matrix
B. A is a zero matrix
C. A is a square matrix
D. None of these Answer


Answer: B
If A is both symmetric and skew-symmetric matrix, then we should have

Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE
Therefore, A is a zero matrix.


Question 15:
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE


Answer
Answer: C
Matrices, NCERT Solutions, Class 12, Maths, Part -5 Notes - JEE

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