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**Definition**

To every square matrix A = [a_{ij}] of order n, we can associate a number (real or complex) called the determinant of the matrix A, written as det A, where a_{ij} is the (i, j)^{th} element of A.

If , then the determinant of A, **denoted by |A| (or det A)**, is given by

Note:

(i) Only square matrices have determinants.

(ii) For a matrix A, A is read as the determinant of A and not, as a modulus of A.

**1. Determinant of a matrix of order one:**

Let A = [a] be the matrix of order 1.

⇒ Determinant of A = a

**2. Determinant of a matrix of order two:**

Let be the matrix of order 2.

⇒ Determinant of A = ad-bc

**3. Determinant of a matrix of order three:**

- The determinant of a matrix of order three can be determined by expressing it in terms of second-order determinants which are known as the expansion of a determinant along a row (or a column).
- Consider the determinant of a square matrix A = [a
_{ij}] 3×3,**i.e.**

Note:In general, if A = kB, where A and B are square matrices of order n,

then |A| = k^{n}|B|, where n = 1, 2, 3.

- We can expand the determinant through any row or column.
**i.e.**Δ = a_{11}A_{11}+ a_{12}A_{12}+ a_{13}A_{13}= a_{11}A_{11}+ a_{21}A_{21}+ a_{31}A_{31}

where A_{ij }is cofactor of a_{ij}.

_{Properties of Determinants}

- If rows are changed into columns and columns into rows, then the values of the determinant remain unaltered.
- If any two-row (or columns) of a determinant are interchanged, the resulting determinant is the negative of the original determinant.
- If two rows (or two columns) in a determinant have corresponding elements that are equal, the value of the determinant is equal to zero.
- If each element in a row (or column) of a determinant is written as the sum of two or more terms then the determinant can be written as the sum of two or more determinants.
- If to each element of a line (row or column) of a determinant be added the equimultiples of the corresponding elements of one or more parallel lines, the determinant remains unaltered.
**i.e.** - If each element in any row (or any column) of the determinant is zero, then the value of the determinant is equal to zero.
- If a determinant D vanishes for x = a, then (x - a) is a factor of D, In other words, if two rows (or two columns) become identical for x = a, then (x-a) is a factor of D.

**Area of triangle**

Assume a triangle with vertices, (x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3}) in a cartisean co-ordinate system as shown below.

Then the area of the given triangle,

It can be represented in the form of the determinant as:

**System of Linear Equation (Cramer's Rule)**

**Case 1: **System of linear equations in two variables.

Let the system of equations be:

a_{1}x + b_{1} y + c_{1} = 0

a_{2}x + b_{2} y + c_{2} = 0, where a_{1}/a_{2} ≠ b_{1}/b_{2}

Solving by cross multiplication, we get,

**Case 2:**** **System of linear equations in three variables.

Let the system of equations be:

a_{1}x + b_{1}y + c_{1}z = d_{1}

a_{2}x + b_{2}y + c_{2}z = d_{2}

a_{3}x + b_{3}y + c_{3}z = d_{3}

Simillarly for y = Δ_{2}/Δ and z = Δ_{3}/Δ

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