## Matrices – Matrix For NCERT Maths Solutions Class 12

12th Mathematics: The arrangement of real numbers in a rectangular array enclosed in brackets as [] or () is known as a Matrix (Matrices is plural of the matrix). Matrix operations are used in electronic physics, computers, budgeting, cost estimation, analysis, and experiments. They are also used in cryptography, modern psychology, genetics, industrial management etc. In general, an m x n matrix is matrix having m rows and n columns. it can be written as follows:

**Order of a Matrix**

There may be any number of rows and any number of columns in a matrix. If there are m rows and n columns in matrix A, its order is m x n and it is read as an m x n matrix.

**Transpose of a Matrix**

The transpose of a given matrix A is formed by interchanging its rows and columns and is denoted by A’.

**Symmetric Matrix**

A square matrix A is said to be a symmetric matrix if A’ = A.

**Skew-Symmetric Matrix**

A square matrix A is said to be a skew-symmetric if A’ = – A. all elements in the principal diagonal of a skew-symmetric matrix are zeroes.

**Addition of Matrix**

If A and B are any two given matrices of the same order, then their sum is defined to be a matrix C whose respective elements are the sum of the corresponding elements of the matrices A and B and we write this as C = A + B.

**Types:**

**Row matrix:** A row matrix has only one row but any number of columns.

**Column matrix:** A column matrix has only one column but any number of rows.

**Square matrix:** A square matrix has the number of columns equal to the number of rows.

**Rectangular Matrix:** A matrix is said to be a rectangular matrix if the number of rows is not equal to the number of columns.

**Diagonal matrix:** a matrix having non-zero elements only in the diagonal running from the upper left to the lower right.

**Scalar Matrix:** A diagonal matrix is said to be a scalar matrix if all the elements in its principal diagonal are equal to some non-zero constant.

**Zero or Null matrix:** If all elements of a matrix are zero, then the matrix is known as zero matrices and denoted by O.

**Unit or Identity matrix:** If in a square matrix has all elements 0 and each diagonal elements are non-zero, it is called identity matrix and denoted by I.

**Equal Matrices:** Two matrices are said to be equal if they are of the same order and if their corresponding elements are equal.

**Properties of Matrix**

- When a matrix is multiplied by a scalar, then each of its elements is multiplied by the same scalar.
- If A and B are any two given matrices of the same order, then their sum is defined to be a matrix C whose respective elements are the sum of the corresponding elements of the matrices A and B and we write this as C = A + B.
- For any two matrices A and B of the same order, A + B = B + A. i.e. matrix addition is commutative.
- For any three matrices A, B and C of the same order, A + (B + C) = (A + B) + C i.e., matrix addition is associative.
- Additive identity is a zero matrix, which when added to a given matrix, gives the same given matrix, i.e., A + O = A = O + A.
- If A + B = O, then the matrix B is called the additive inverse of the matrix of A.
- If A and B are two matrices of order m x p and p x n respectively, then their product will be a matrix C of order m x n.

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**Invertible Matrix**

A square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = I = BA, Where I is identity matrix of order n.

**Theorems of invertible matrices**

*Theorem 1:*Every invertible matrix possesses a unique inverse.*Theorem 2:*A square matrix is invertible if it is non-singular.

Example. Construct a 3 × 2 matrix whose elements are given by a_{ij}= 1/2|i-3j|

Solution. In general, a 3 × 2 matrix is given by A=

Now a_{ij}= ½ |i-3j| ,i = 1, 2, 3 and j = 1, 2.

Therefore a_{11}= ½ |1-3*1|a_{12}= ½|1-3*2| = 5/2

a_{21}= ½ |2-3*1| = 1/2 a_{22} = ½ |2-3*2| = 2

a_{31} = ½ |3-3*1| = 0 a_{32 }= ½ |3-3*2| = 3/2

Hence the required matrix is given by A=

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