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Matrices - Ex 2.1 | Maharashtra Board 12th Maths Solutions Chapter 2

Matrices - Ex 2.1 | Maharashtra Board 12th Maths Solutions Chapter 2

Maharashtra Board 12th Maths Solutions Chapter 1 Mathematical Logic Ex 1.1

Question 1.Apply the given elementary transformation on each of the following matrices.

1) A = [1103], R1 ↔ R2
Solution:
A = [1103]
By R1 ↔ R2, we get,
A ~ [1130]

Question 2.
B = [121534], R1 → R1 → R2
Solution:
B = [121534],
R1 → R1 → R2 gives,
B ~ [126514]

Question 3.
A = [5143], C1 ↔ C2; B = [3415], R1 ↔ R2. What do you observe?
Solution:
A = [5143]
By C1 ↔ C2, we get,
A ~ [4351] …(1)
B = [3415]
By R1 ↔ R2, we get,
B ~ [4351] …(2)
From (1) and (2), we observe that the new matrices are equal.

Question 4.
A = [102113], 2C2
B = [120425], -3R1


Find the addition of the two new matrices.
Solution:

A = [102113]
By 2C2, we get,
A ~ [104213]
B = [120425]
By -3R1, we get,
B ~ [320465]
Now, addition of the two new matrices



Question 5.
A = 123113301, 3R3 and then C3 + 2C2.
Solution:
A = 123113301
By 3R3, we get
A ~ 129119303
By C3 + 2C2, we get,
A ~ 1291193+2(1)0+2(1)3+2(9)
∴ A ~ 1291191221

Question 6.
A = 123113301, C3 + 2C2 and then 3R3. What do you conclude from Ex. 5 and Ex. 6 ?
Solution:

A = 123113301
By C3 + 2C2, we get,
A ~ 1231133+2(1)0+2(1)1+2(3)
∴ A ~ 123113127
By 3R3, we get
A ~ 1291191221
We conclude from Ex. 5 and Ex. 6 that the matrix remains same by interchanging the order of the elementary transformations. Hence, the transformations are commutative.

Question 7.
Use suitable transformation on [1324] into an upper triangular matrix.
Solution:
Let A = [1324]
By R2 – 3R1, we get,
A ~ [1022]
This is an upper triangular matrix.

Question 8.
Convert [1213] into an identity matrix by suitable row transformations.
Solution:
Let A = [1213]
By R2 – 2R1, we get,
A ~ [1015]
By (15)R2, we get,
A ~ [1011]
By R1 + R2, we get,
A ~ [1001]
This is an identity matrix.

Question 9.
Transform 123112234 into an upper triangular matrix by suitable row
transformations.
Solution:
Let A = 123112234
By R2 – 2R1 and R3 – 3R1, we get
A ~ 100135212
By R3 – (53)R2, we get,
A ~ 1001302113
This is an upper triangular matrix.


Matrices - Ex 2.1 | Maharashtra Board 12th Maths Solutions Chapter 2


A matrix of order mxm is a square arrangement of m2 elements. The corresponding determinant of the same elements, after expansion is seen to be a value which is an element itself. In standard XI, we have studies the types of matrices and algebra of matrices namely addition, subtraction, multiplication of two matrices. 

The matrices are useful in almost every branch of science. Many problems in Statistics are expressed in terms of matrices. Matrices are also useful in Economics, Operation Research. It would not be an exaggeration to say that the matrices are the language of atomic Physics. Hence, it is necessary to learn the uses of matrices with the help of elementary transformations and the inverse of a matrix

Definition of Elementary Transformation of Matrix in Maths

2.1 Elementary Transformation : Let us first understand the meaning and applications of elementary transformations. The elementary transformation of a matrix are the six operations, three of which are due to row and three are due to column. They are as follows :

How do you interchange rows and columns of a matrix?

(a) Interchange of any two rows or any two columns. If we interchange the ith row and the jth row of a matrix then after this interchange the original matrix is transformed to a new matrix. This transformation is symbolically denoted as Ri ↔ Rj or Rij. The similar transformation can be due to two columns say Ck ↔ Ci or Cki. (Recall that R and C symbolically represent the rows and columns of a matrix.)

What is the inverse matrix 

2.2 Inverse of a matrix : Definition : If A is a square matrix of order m and if there exists another square matrix B of the same order such that AB = BA = I, where I is the identity matrix of order m, then B is called as the inverse of A and is denoted by A–1. Using the notation A-1 for B we get the above equation as AA–1 = A–1A = I. Hence, using the same definition we can say that A is also the inverse of B.

Note that - 
(1) Every square matrix A of order m × m has its corresponding determinant; det A = |A| 
(2) A matrix is said to be invertible if its inverse exists. 
(3) A square matrix A has inverse if and only if |A| ≠ 0 Uniqueness of inverse of a matrix It can be proved that if A is a square matrix where |A| ≠ 0 then its inverse, say A-1, is unique. 

Theorem : Prove that if A is a square matrix and its inverse exists then it is unique. 
Proof : Let, ‘A’ be a square matrix of order ‘m’ and let its inverse exist. Let, if possible, B and C be the two inverses of A. Therefore, by definition of inverse AB = BA = I and AC = CA = I.

2.2.1 Inverse of a nonsingular matrix by elementary transformation : 

By definition of inverse of A, if A–1 exists then AA-1 = A-1 A = I. Let us consider the equation AA-1 = I. Here A is the given matrix of order m and I is the identity matrix of order ‘m’. Hence the only unknown matrix is A-1. Therefore, to find A-1, we have to first convert A into I. This can be done by using elementary transformations. Here we note that whenever any elementary row transformation is to be applied on the product AB = C of two matrices A and B, it is enough to apply it only on the prefactor, A. B remains unchanged. And apply the same row transformation to C


Hence, the equation AA–1 = I can be transformed into an equation of the type IA–1 = B, by applying same series of row transformations on both the sides of the equation. However, if we start with the equation A-1A = I (which is also true by the definition of inverse) then the transformation of A should be due to the column transformation. Apply column transformation to post factor and other side, where as prefactor remains unchanged. Thus, starting with the equation AA–1 = I , we perform a series of row transformations on both sides of the equation, so that ‘A’ gets transformed to I. Thus,


Inverse of a square matrix by adjoint method : 

From the previous discussion about finding the inverse of a square matrix by elementary transformation it is clear that the method is elaborate and requires a series of transformations. There is another method for finding the inverse and it is called as the inverse by the adjoint method. This method can be directly used for finding the inverse. However, for understanding this method you should know the definition of a minor, a co-factor and adjoint of the given matrix.


Let us first recall the definition of minor and co-factor of an element of a determinant. 

Definition : Minor of an element aij of a determinant is the determinant obtained by deleting i th row and j th column in which the element aij lies. Minor of an element aij is denoted by Mij. (Can you find the order of the minor of any element of a determinant of order ‘n’?)

Definition : Co-factor of an element aij of a determinant is given by (-1)i + j Mij , where Mij is minor of the element aij. Co-factor of an element aij is denoted by Aij. Now for defining the adjoint of a matrix, we require the co-factors of the elements of the matrix. 

Matrices - Ex 2.1 | Maharashtra Board 12th Maths Solutions Chapter 2

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Matrices - Ex 2.1 | Maharashtra Board 12th Maths Solutions Chapter 2

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