Miscellaneous

Chapter 4 Class 12 Determinants
Serial order wise

Let’s look at various properties of Matrices and Determinants

## Addition and Subtraction of Matrices

• A + B = B + A
• (A + B) + C = A + (B + C)
• k (A + B) = kA + kB

## Multiplication of matrices

• AB ≠ BA
• (AB) C = A (BC)
• Distributive law
A (B + C) = AB + AC
(A + B) C = AC + BC
• Multiplicative identity
For a square matrix A
AI = IA = A

## Properties of transpose of matrix

• (A T ) T = A
• (kA) T = kA T
• (A + B) T = A T + B T
• (AB) T = B T A T

## Symmetric and Skew Symmetric matrices

• Symmetric Matrix - If A T = A

• Skew - symmetric Matrix - If A T = A
Note: In a skew matrix, diagonal elements are always 0 .

• For any square matrix A,
(A + A T ) is a symmetric matrix
(A − A T ) is a skew-symmetric matrix

## Inverse of a matrix

For a square matrix A, if

AB = BA = I

Then, B is the inverse of A

i.e. B = A −1

We will find inverse of a matrix by

## Properties of Inverse

1. For a matrix A,
A −1 is unique, i.e., there is only one inverse of a matrix

2. (A −1 ) −1 = A

3. (𝑘 𝐴) −1 = 1/𝑘 𝐴 −1
Note: This is different from
(kA) T = k A T
4. (A -1 ) T = (A T ) -1

5. (A + B) -1 = A -1 + B -1

6. (𝐴𝐵) −1 = 𝐵 −1 𝐴 −1

## Important things to note in Determinants

1. Determinant of Identity matrix = 1
det (I) = 1

2. |A T | = |A|

3. |AB| = |A| |B|

4. |A −1 | = 1/|𝐴|

5. |kA| = k n |A| where n is order of matrix

6. Similarly,
|−A| = |−1 × A|
= (−1) n × |A|

7. (adj A) A = A (adj) = |A|I

8. Deteminant of adj A
|adj A| = |A| 𝑛−1
where n is the order of determinant

## Other important points

Also, look at

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### Transcript

Number multiplied to matrix and determinant Matrix If a number is multiplied to matrix, it is multiplied to each element of the matrix 2 [■8(9&2&[email protected]&−1&[email protected]&0&−2)] = [■8(2×9&2×2&2×[email protected]×5&2×(−1)&2×[email protected]×4&2×0&2×(−2))] Determinant If a number is multiplied to determinant, it is multiplied to either one row, or one column 2 |■8(9&2&[email protected]&−1&[email protected]&0&−2)| = |■8(2×9&2×2&2×[email protected]&−1&[email protected]&0&−2)| Or |■8(2×9&2&[email protected]×5&−1&[email protected]×4&0&−2)|

Made by

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.