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

1) A = **Solution**:

A =

By R1 ↔ R2, we get,

A ~

**Question 2.**

B =

Solution:

B =

R1 → R1 → R2 gives,

B ~

**Question 3.**A =

Solution:

A =

By C1 ↔ C2, we get,

A ~

B =

By R1 ↔ R2, we get,

B ~

From (1) and (2), we observe that the new matrices are equal.

**Question 4.**

A =

B =

**Find the addition of the two new matrices.Solution:**

A =

By 2C2, we get,

A ~

B =

By -3R1, we get,

B ~

Now, addition of the two new matrices

**Question 5.**

A =

Solution:

A =

By 3R3, we get

A ~

By C3 + 2C2, we get,

A ~

∴ A ~

**Question 6.**

A = **Ex. 6 ?Solution:**

A =

By C3 + 2C2, we get,

A ~

∴ A ~

By 3R3, we get

A ~

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

**Solution**:

Let A =

By R2 – 3R1, we get,

A ~

This is an upper triangular matrix.

**Question 8.****Convert [12−13] into an identity matrix by suitable row transformations.**

Solution:

Let A =

By R2 – 2R1, we get,

A ~

By

A ~

By R1 + R2, we get,

A ~

This is an identity matrix.

**Question 9.Transform ⎡⎣⎢123−112234⎤⎦⎥ into an upper triangular matrix by suitable row** transformations.

Solution:

Let A =

By R2 – 2R1 and R3 – 3R1, we get

A ~

By R3 –

A ~

This is an upper triangular matrix.

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

### 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?

### 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.

**2.2.1 Inverse of a nonsingular matrix by elementary transformation : **

### Inverse of a square matrix by adjoint method :

**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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