Any rectangular arrangement of numbers in m rows and n columns is called a matrix of order m×n. Matrices and determinants is an important topic for the JEE exam. These formulas will help students to have a quick revision before the exam. Students can expect 2-3 questions from this topic.

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Matrices and Determinants Formulas

Matrices

Any rectangular arrangement of numbers in m rows and n columns is called a matrix of order m×n.

\begin{array}{l} A = [\begin{array}{c} a_{11} & a_{12} & a_{13} & \dots & a_{1 j} & \dots . & a_{1 n} \\ a_{21} & a_{22} & a_{23} & \dots & a_{2 j} & \dots . & a_{2 n} \\ \dots & \dots & \dots & \dots & \dots & \dots . & \dots \\ a_{m 1} & a_{m 2} & a_{m 3} & \dots & a_{m j} & \dots . & a_{m n} \end{array}] \end{array}

Where a_ijdenotes the element of the i^throw and j^th column. The above matrix is denoted as [a_ij]_m×n . The elements a₁₁, a₂₂, a₃₃ etc are called diagonal elements. Their sum is called the trace of A denoted by T_r(A).

2. Basic Definitions

(i) Row matrix: A matrix having one row is called a row matrix.

(ii) Column matrix: A matrix having one column is called a column matrix.

(iii) Square matrix: A matrix of order m×n is called square matrix if m = n.

(iv) Zero matrix: A = [a_ij]_m×n is called a zero matrix, if a_ij = 0 for all i and j.

(v) Upper triangular matrix: A = [a_ij]_m×n is said to be upper triangular, if a_ij= 0 for i > j.

(vi) Lower triangular matrix: A = [a_ij]_m×n is said to be lower triangular, if a_ij = 0 for i < j.

(vii) Diagonal matrix: A square matrix [a_ij]_m×n is said to be diagonal, if a_ij = 0 for i ≠ j.

(viii) Scalar matrix: A diagonal matrix A = [a_ij]_m×n is said to be scalar, if a_ij = k for i = j.

(ix) Unit matrix (Identity matrix): A diagonal matrix A = [a_ij]_n is a unit matrix, if a_ij = 1 for i = j.

(x) Comparable matrices: Two matrices A and B are comparable, if they have the same order.

3. Equality of matrices: Two matrices A = [a_ij]_m×n and B = [b_ij]_p×q are are said to be equal, if m = p and n = q and a_ij = b_ij ∀ i and j.

4. Multiplication of a matrix by a scalar: Let λ be a scalar, then λA = [b_ij]_m×n where b_ij= λa_ij ∀ i and j.

5. Addition of matrices: Let A = [a_ij]_m×n and B = [b_ij]_m×n be two matrices, then A+B = [a_ij]_m×n+ [b_ij]_m×n = [c_ij]_m×n where c_ij = a_ij+b_ij∀ i and j.

6. Subtraction of matrices: A-B = A+(-B), where -B = ( -1)B.

7. Properties of addition and scalar multiplication:

(i) λ(A+B) = λA+λB

(ii) λA = Aλ

(iii) (λ₁+λ₂)A = λ₁A+λ₂A

8. Multiplication of matrices: Let A = [a_ij]_m×p and B = [b_ij]_p×n , then AB = [c_ij]_m×nwhere

\begin{array}{l} c_{i j} = \sum_{k = 1}^{p} a_{i k} b_{k j} \end{array}

.

9. Properties of matrix multiplication:

(i) AB ≠ BA

(ii) (AB)C = A(BC)

(iii) AI_n = A = I_nA

(iv) For every non singular square matrix A (i.e., | A |≠ 0 ) there exists a unique matrix B so that AB = I_n = BA. In this case we say that A and B are multiplicative inverses of one another. I.e., B = A^-1 or A = B^-1 .

10. Transpose of a Matrix.

Let A = [a_ij]_m×n then A^’or A^Tthe transpose of A is defined as A’ = [a_ji]_n×m .

(i) (A’)’ = A

(ii) (λA)’ = λA’

(iii) (A+B)’ = A’+B’

(iv) (A-B)’ = A’-B’

(v) (AB)’ = A’B’

(vi) For a square matrix A, if A’ = A , then A is said to be a symmetric matrix.

(vii) For a square matrix A, if A’ = -A , then A is said to be a skew symmetric matrix.

11. Submatrix of a matrix:

Let A be a given matrix. The matrix obtained by deleting some rows and columns of A is called a submatrix of A.

12. Properties of determinant:

(i) | A | = | A’ | for any square matrix A.

(ii) If two rows or two columns are identical, then | A | = 0.

(iii) If | λA | = λⁿ| A |, when A = [a_ij]_n×n.

(iv) If A and B are two square matrices of the same order, then | AB |= | A || B |

13. Singular and Non-singular matrix:

A square matrix A is said to be singular, if | A | = 0.

A square matrix A is said to be non-singular, if | A | ≠ 0.

14. Cofactor and adjoint matrix.

Let A = [a_ij]_n×nbe a square matrix. The matrix obtained by replacing each element of A by the corresponding cofactor is called the cofactor matrix of A. The transpose of the matrix of the cofactor of A is called the adjoint of A, denoted as adj A.

15. Properties of adj A.

(i) A . adj A = | A |I_n = (adj A)A where A = [a_ij]_n×n.

(ii) | adj A | = | A |^n-1 , where n is the order of A.

(iii) If A is a symmetric matrix, then adj A is also symmetric.

(iv) If A is singular, then adj A is also singular.

(v) Let A be non singular matrix, then

\begin{array}{l} \frac{a d j A}{| A |} is the multiplicative inverse of A and is denoted by A^{- 1} . \end{array}

(vi) (A^-1)^T = (A^T)^-1 for any non singular matrix.

(vii) (A^-1)^-1 = A, if A is non singular.

(viii) A^-1 is always non singular.

(ix) (adj A^T) = (adj A)^T

(x) Let k be a non zero scalar and A be a non singular matrix, then

\begin{array}{l} (k A)^{- 1} = \frac{1}{k} A^{- 1} \end{array}

(xi) | A^-1 | = 1/|A| for | A | ≠ 0.

(xii) Let A be a non singular matrix, then AB = AC⇒ B = C and BA = CA ⇒ B = C.

16. System of linear equations and matrices:

System of linear equations AX = B is said to be consistent if it has at least one solution.

(i) System of linear equations and matrix inverse:

(a) If A is non-singular, solution is given by X = A^-1B.

(b) If A is singular, adj(A) B = 0 and no two columns of A are proportional, then the system has infinitely many solutions.

(c) If A is singular and (adj A)B ≠ 0, then the system has no solution.

(ii) Homogeneous system and matrix inverse:

If the above system is homogeneous, n equations in n unknowns, then in the matrix form it is AX = 0. ( since b₁ = b₂ =….. b_n = 0), where A is a square matrix.

If A is non-singular, the system has only one trivial solution. X = 0.

If A is singular, then the system has infinitely many solutions (including the trivial solution) and hence it has non-trivial solutions.

(iii) Elementary row transformation of Matrix:

The following operations on a matrix are called elementary row transformations.

(a) Interchanging two rows.
(b) Multiplication of all the elements of a row by a non zero scalar.

(c) Addition of a constant multiple of a row to another row.

17. Characteristic Polynomial and characteristic equation.

Let A be a square matrix, then the polynomial | A-xI | is called the characteristic polynomial of A and the equation | A-xI | = 0 is called the characteristic equation of A.

18. Cayley Hamilton theorem:

Every square matrix A satisfies its characteristic equation. I.e., a₀xⁿ+ a₁x^n-1+ ….. + a_n-1x + a_n = 0 is the characteristic equation of A, then a₀Aⁿ+ a₁A^n-1+ ……..+ a_n-1A + a_nI = 0

19. More definitions on matrices:

(i) Nilpotent matrix: A square matrix A is called nilpotent if A^p = 0 for some positive integer. If p is the smallest such positive integer, then p is called its nilpotency.

(ii) Idempotent matrix: A square matrix A is said to be idempotent if, A² = A.

(iii) involutory matrix: A square matrix A is said to be involutory if, A² = I.

(iv) Orthogonal matrix: A square matrix A is said to be orthogonal if, A^TA = I = AA^T.

(v) Unitary matrix: A square matrix A is said to be unitary if

\(\begin{array}{l}A\: (\bar{A})^{T}= I,\ \text{where}\ \bar{A}\ \text{

is the complex conjugate of A.}\end{array} \)

Determinants

We write the expression a₁b₂– a₂b₁ as

\begin{array}{l} | \begin{array}{c} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array} | \end{array}

and

\begin{array}{l} | \begin{array}{c} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array} | is called a determinant of order 2. \end{array}

1. Expansion of determinant:

\begin{array}{l} | \begin{array}{c} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} | \end{array}

is called the determinant of order three. Its value can be found as

D = a₁(b₂c₃– b₃c₂) – b₁(a₂c₃ – a₃c₂) + c₁(a₂b₃ – a₃b₂)

2. Minors: The minor of a_ij is obtained by deleting i^throw and j^thcolumn from the determinant. It is denoted by M_ij.

3. Cofactor: Cofactor of element a_ij is C_ij = (-1)^i+j M_ij

D = a₁₁M₁₁– a₁₂M₁₂+ a₁₃M₁₃ = a₁₁C₁₁+ a₁₂C₁₂+ a₁₃C₁₃

4. Transpose of a Determinant: The transpose of a determinant is a determinant obtained after interchanging the rows and columns.

\begin{array}{l} D = | \begin{array}{c} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} | \end{array}

and

\begin{array}{l} D^{T} = | \begin{array}{c} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array} | \end{array}

5. Symmetric, Skew symmetric, Asymmetric Determinants:

(i) A determinant is symmetric if it is identical to its transpose. The i^throw is identical to its i^thcolumn, i.e. a_ij = a_ji for all values of i and j.

(ii) A determinant is skew-symmetric if it is identical to its transpose, having the sign of each element inverted, i.e. a_ij = -a_ji for all values of i and j.

(iii) A determinant is asymmetric if it is neither symmetric nor skew-symmetric.

6. Properties of determinants:

(i) D = D’

(ii) If a determinant has all the elements zero in any row or column, then D = 0

(iii) If any two rows or columns of a determinant be interchanged, then D’ = -D.

(iv) If a determinant has any two rows or columns identical, then D = 0.

(v) If all the elements of any row or column be multiplied by the same number k, then D’ = kD.

(vi) If each element of any row or column can be expressed as a sum of two terms, then the determinant can be expressed as the sum of two determinants. i.e.

\begin{array}{l} | \begin{array}{c} a_{1} + x & b_{1} + y & c_{1} + z \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} | = | \begin{array}{c} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} | + | \begin{array}{c} x & y & z \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} | \end{array}

(vii) The value of a determinant is not altered by adding to the elements of any row or column a constant multiple of the corresponding elements of any other row or column.

7. Multiplication of two determinants:

\begin{array}{l} | \begin{array}{c} _{} a_{1} & b_{1} \\ a_{2} & b_{2} \end{array} | \times | \begin{array}{c} l_{1} & m_{1} \\ l_{2} & m_{2} \end{array} | = | \begin{array}{c} a_{1} l_{1} + b_{1} l_{2} & a_{1} m_{1} + b_{1} m_{2} \\ a_{2} l_{1} + b_{2} l_{2} & a_{2} m_{1} + b_{2} m_{2} \end{array} | \end{array}

8. Summation of determinants:

Let

\begin{array}{l} Δ (r) = | \begin{array}{c} f (r) & g (r) & h (r) \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{array} | \end{array}

, then

\begin{array}{l} \sum_{r = 1}^{n} Δ (r) = | \begin{array}{c} \sum_{r = 1}^{n} f (r) & \sum_{r = 1}^{n} g (r) & \sum_{r = 1}^{n} h (r) \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{array} | \end{array}

Where a₁, a₂, a₃, b₁, b₂, and b₃ are constants independent of r.

9. Integration of a determinant:

Let

\begin{array}{l} Δ (x) = | \begin{array}{c} f (x) & g (x) & h (x) \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array} | \end{array}

,

then

\begin{array}{l} \int_{a}^{b} Δ (x) d x = | \begin{array}{c} \int_{a}^{b} f (x) & \int_{a}^{b} g (x) & \int_{a}^{b} h (x) \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array} | \end{array}

Where a₁, a₂, a₃, b₁, b₂, b₃ are constants independent of x.

10. Differentiation of determinants:

Let

\begin{array}{l} Δ (x) = | \begin{array}{c} f_{1} (x) & f_{2} (x) & f_{3} (x) \\ g_{1} (x) & g_{2} (x) & g_{3} (x) \\ h_{1} (x) & h_{2} (x) & h_{3} (x) \end{array} | \end{array}

then,

\begin{array}{l} Δ ‘ (x) = | \begin{array}{c} f_{1}^{'} (x) & f_{2}^{'} (x) & f_{3}^{'} (x) \\ g_{1} (x) & g_{2} (x) & g_{3} (x) \\ h_{1} (x) & h_{2} (x) & h_{3} (x) \end{array} | + | \begin{array}{c} f_{1} (x) & f_{2} (x) & f_{3} (x) \\ g_{1}^{'} (x) & g_{2}^{'} (x) & g_{3}^{'} (x) \\ h_{1} (x) & h_{2} (x) & h_{3} (x) \end{array} | + | \begin{array}{c} f_{1} (x) & f_{2} (x) & f_{3} (x) \\ g_{1} (x) & g_{2} (x) & g_{3} (x) \\ h_{1}^{'} (x) & h_{2}^{'} (x) & h_{3}^{'} (x) \end{array} | \end{array}

11. Cramer’s rule:

(i) Two variables:

(a) Consistent equations: Definite and unique solution. [Intersecting lines]

(b) Inconsistent equation: No solution. [Parallel line]

(c) Dependent equation: Infinite solutions. [Identical lines]

Let a₁x+b₁y+c₁ = 0 and a₂x+b₂y+c₂ = 0 then,

\begin{array}{l} \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} \end{array}

⇒ given equations are inconsistent.

If

\begin{array}{l} \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \end{array}

⇒ given equations are dependent.

(ii) Three variables:

Let a₁x + b₁y + c₁z = d₁ , a₂x + b₂y + c₂z = d₂ and a₃x + b₃y + c₃z = d₃, then x = D₁/D, y = D₂/D and z = D₃/D, where

\begin{array}{l} D = | \begin{array}{c} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} | \end{array}

;

\begin{array}{l} D_{1} = | \begin{array}{c} d_{1} & b_{1} & c_{1} \\ d_{2} & b_{2} & c_{2} \\ d_{3} & b_{3} & c_{3} \end{array} | \end{array}

;

\begin{array}{l} D_{2} = | \begin{array}{c} a_{1} & d_{1} & c_{1} \\ a_{2} & d_{2} & c_{2} \\ a_{3} & d_{3} & c_{3} \end{array} | \end{array}

;

\begin{array}{l} D_{3} = | \begin{array}{c} a_{1} & b_{1} & d_{1} \\ a_{2} & b_{2} & d_{2} \\ a_{3} & b_{3} & d_{3} \end{array} | \end{array}

(iii) Consistency of a system of equations:

(a) If D ≠ 0 and at least one of D₁, D₂, D₃ ≠ 0, then the given system of equations is consistent and has a unique non-trivial solution.

(b) If D ≠ 0 and D₁ = D₂ = D₃ = 0, then the given system of equations are consistent and has trivial solution only.

(c) If D = D₁ = D₂ = D₃ = 0, then the given system of equations have either infinite solutions or no solution.

(d) If D = 0 but atleast one of D₁, D₂, or D₃ is not zero, then the equations are inconsistent and have no solution.

(e) If a given system of linear equations has only zero solution for all its variables, then the given equations are said to have trivial solution.

(iv) Three equation in two variable:

If x and y are not zero, then the condition for a₁x + b₁y + c₁ = 0; a₂x + b₂y + c₂ = 0 and a₃x + b₃y + c₃ = 0 to be consistent in x and y is

\begin{array}{l} | \begin{array}{c} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} | = 0 \end{array}

13. Application of Determinants:

(i) Area of a triangle whose vertices are (x_r, y_r) ;r = 1, 2, 3 is

\begin{array}{l} D = \frac{1}{2} | \begin{array}{c} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array} | \end{array}

. If D = 0, then the three points are collinear.

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