Matrices

A matrix is defined as a rectangular array of elements. If the arrangement has m rows and n columns, then the matrix is of order m × n (read as m by n).

A matrix is enclosed by a pair of parameters such as ( ) or [ ]. It is denoted by a capital letter.

• Two matrices are said to be comparable if they have the same order.
• Addition and subtraction of two matrices is possible only if they have the same order.
• If two matrices A and B are of same order, then A - B = A + (- B).
• Commutative law, associative law holds good for addition of matrices.
• The additive identity of a matrix A of order m×n is the zero matrix of order m×n.
• The additive inverse of a matrix A is -A.
• The multiplication of two matrices A and B is possible if the number of columns of A is equal to the number of rows B.
• Suppose A is a matrix of order m×n and B is a matrix of order n×p, the matrix AB is of order m×p.

• If A, B and C are the matrices which can be multiplied then

(a)Matrix multiplication is not commutative,

i.e., AB BA (always)

(b) Associative law holds good for matrix multiplication,

i.e., (AB)C = A(BC)

(c) Matrix multiplication is distributive with respect to addition

A(B + C) = AB + AC

or (A + B)C = AC + BC

• If A is a matrix of order m×n and is a scalar (real or complex) then the matrix kA is obtained by multiplying each element of the matrix A by k.

• To every square matrix, a value can be associated which is known as the determinant of the matrix.

Note that the determinant of kA where k is a scalar and A is a square matrix, is given by kⁿ times determinant of A.

i.e., is |kA| = kⁿ |A|

• The value of the determinant remain unchanged if its rows and columns are interchanged

• If two rows or columns of a determinant are interchanged, then the sign of the determinant is changed.

• If any two rows or columns of a determinant are equal, then its value is zero.

• If each element of a row or column of a determinant multiplied by k, then its value is multiplied by k.

• If two rows or columns of determinant are proportional, the value of the determinant is zero.

• A square A = [a_ij] is said to be symmetric if A^T = A, i.e., if

a_ij = a_ji

• A square matrix A is said skew symmetric if A^T = - A,

i.e., a_ij = - a_ji

• Any square matrix A can be expressed as the sum of a symmetric matrix and a skew symmetric matrix as follows

• For a 2 x 2 matrix, the adjoint is got by interchanging elements in the leading diagonal and changing signs in the other diagonal.

⇒ The value of the determinant `|[a,b],[c,d]|` is ad-bc

• If A =[a_ij]_mxn is a matrix of order m×n. The minor of a_ij of |A|, denoted by M_ij, is given by the determinant which is obtained by deleting i^th row j^th column of |A|.

• The co-factor of the determinant of the A = [a^ij]_mxn, denoted by A_ij is given by A_ij = (-1)^i+j M_ij

• The transpose of a matrix A, denoted by A^T, is obtained by interchanging the rows and columns of A.

• The adjoint of a square matrix A = [a_ij] is defined as the transpose of the matrix [A_ij] where A_ij is the co-factor of the element a_ij.

Adjoint of A is denoted by Adj A.

Note that the concept of adj is only for square matrix.

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

• Let A be a square matrix of order n. If there exists a square matrix B of order n, such that AB = BA = I_n, where I_n is the identify matrix of order n, then B is called the inverse of A.

• The inverse of a matrix A exists if and only if |A| 0.

In other words, every non-singular matrix is invertible.

• The area of a triangle whose vertices are (x₁, y₁), (x₂, y₂) and (x3, y3), is given by the determinant `|[x_1,y_2,1],[x_2,y_2,1],[x_3,y_3,1]|` . Solve a system of Linear Equation using Cramer's Rule.

The following are the steps to solve a system of linear equations

a₁x + b₁y + c₁ = d₁
a₂x + b₂y + c₂ = d₂
a₃x + b₂y + c₃ = d₃

using Cramer's rule.

Step 1: Find the value of the determinant

D = `|[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]|`

Step 2: If D 0, then the system has unique solution, given by

x = $\frac{D_1}{D}$ y = $\frac{D_2}{D}$ z = $\frac{D_3}{D}$

Where D₁, D₂ and D₃ are the determinants obtained from D by replacing respectively the first column, 2^nd column and third column containing the constant terms d₁, d₂, d₃.

Step 3: If D = 0, the system may have infinite number of solutions or no solution.

Let A be a square matrix of order n. Following are the steps to find the inverse of a matrix.

Step 1: Find the value of the determinants A. That is, find |A|

Step 2: If |A| = 0, inverse of the matrix A does not exists.

Step 3: If |A| 0, find the co-factors A_ij of all the elements of A.

Step 4: Find adj A, the transpose of the matrix of co-factors A_ij.

Step 5:

Following are the steps to solve a system of linear equations with three unknown, using inverse of a matrix (Matrix method)

Let the given system of equations be

a₁x + b₁y + c₁ = d₁
a₂x + b₂y + c₂ = d₂
a₃x + b₂y + c₃ = d₃

Step 1: A = `|[a_1,b_1,c_1],[a_2,b_2,c_2],[a_3,b_3,c_3]|` , X =` [[x],[y],[z]]` , B = `[[d_1],[d_2],[d_3]]`

The system of linear equations may be expressed as AX = B.

Step 2: Find |A|. If |A| 0, the system has unique solution which is given by X = A^-1B.

Step 3:

If |A| = 0, put x = k (y = k or z = k) in any two of the given equations and find y and z in terms of k.

Substitute these values of x, y and z in terms of k in the third equation. If the third equation is satisfied by these values of x, y and z, then the system has infinitely many solutions.

If the third equation is not satisfied, the system has no solution.