# Mathematics > Matrices and Determinants

### Matrices

• A matrix is an ordered rectangular array of numbers or functions.
• A matrix having m rows and n columns is called a matrix of order m × n.
•  is a column matrix.
• is a row matrix.
• An m × n matrix is a square matrix if m = n.
• A =  is a diagonal matrix if  = 0, when i ≠ j.
• A =  is a scalar matrix if   = 0, when i ≠ j,   = k, (k is some constant), when i = j.
• A =  is an identity matrix, if  = 1, when i = j,  = 0, when i ≠ j.
• A zero matrix has all its elements as zero.
• A = = B if (i) A and B are of same order, (ii)  for all possible values of i and j.
• – A = (–1)A
• A – B = A + (–1) B
• A + B = B + A
• (A + B) + C = A + (B + C), where A, B and C are of same order.
• k(A + B) = kA + kB, where A and B are of same order, k is constant.
• (k + l ) A = kA + lA, where k and l are constant.
• If A =  and , then  (i) A(BC) = (AB)C, (ii) A(B + C) = AB + AC, (iii) (A + B)C = AC + BC
• If A = [aij]m × n, then A` or AT = (i) (A`)` = A, (ii) (kA)` = kA`, (iii) (A + B)` = A` + B`, (iv) (AB)` = B`A`
• A is a symmetric matrix if A` = A.
• A is a skew symmetric matrix if A` = –A.
• Any square matrix can be represented as the sum of a symmetric and a skew symmetric matrix.
• Elementary operations of a matrix are as follows:  (i)
• If A and B are two square matrices such that AB = BA = I, then B is the inverse matrix of A and is denoted by A–1 and A is the inverse of B.
• Inverse of a square matrix, if it exists, is unique.

### Determinants

• Determinant of a matrix A =  is given by .
• For any square matrix A, the |A| satisfy following properties.
• |A`| = |A|, where A` = transpose of A.
• If we interchange any two rows (or columns), then sign of determinant changes.
• If any two rows or any two columns are identical or proportional, then value of determinant is zero.
• If we multiply each element of a row or a column of a determinant by constant k, then value of determinant is multiplied by k.
• Multiplying a determinant by k means multiply elements of only one row (or one column) by k.
• If  A =  , then k .A =k3 A
• If elements of a row or a column in a determinant can be expressed as sum of two or more elements, then the given determinant can be expressed as sum of two or more determinants.
• If to each element of a row or a column of a determinant the equimultiples of corresponding elements of other rows or columns are added, then value of determinant remains same.
• Minor of an element aij of the determinant of matrix A is the determinant obtained by deleting ith row and jth column and denoted by.
• Cofactor of of given by  = (– 1)i+ j
• Value of determinant of a matrix A is obtained by sum of product of elements of a row (or a column) with corresponding cofactors. For example,
• A =
• If elements of one row (or column) are multiplied with cofactors of elements of any other row (or column), then their sum is zero. For example,  =0
• A (adj A) = (adj A) A = |A| I, where A is square matrix of order n.
• A square matrix A is said to be singular or non-singular according as |A| = 0 or |A| ≠ 0.
• If AB = BA = I, where B is square matrix, then B is called inverse of A.
• Also A–1 = B or B–1 = A and hence (A–1)–1 = A.
• A square matrix A has inverse if and only if A is non-singular.
• A-1 = 1/|A|
• Unique solution of equation AX = B is given by X = A–1 B, where A ≠ 0.
• A system of equation is consistent or inconsistent according as its solution exists or not.
• For a square matrix A in matrix equation AX = B
1. |A| ≠ 0, there exists unique solution\
2. |A| = 0 and (adj A) B ≠ 0, then there exists no solution
3. |A| = 0 and (adj A) B = 0, then system may or may not be consistent.

Area of a triangle with vertices is given by