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**NCERT mathematics class 12 chapter Determinants Chapter Summary**. You can find the summary for the**chapter 4**of**NCERT class 12 mathematics**in this page. So is the case if you are looking for**NCERT class 12 Maths**related topic**Matrices**. This page contains summary of the chapter. If you’re looking for exercise solutions, they’re available at●

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Determinants – Summary

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Determinant of a matrix {\text{A} = {[a_{11}]}_{1 × 1}} is given by {\left|a_{11}\right| = a_{11}}

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Determinant of a matrix {\text{A} = \left[\begin{array}{cc} a_{11} & a_{12} \\[5pt] a_{21} & a_{22} \end{array}\right]} is given by

{\left|\text{A}\right| = \left|\begin{array}{cc} a_{11} & a_{12} \\[5pt] a_{21} & a_{22} \end{array}\right| = a_{11}a_{22} - a_{12}a_{21}}

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Determinant of a matrix

{\text{A} = \left[\begin{array}{cc} a_1 & b_1 & c_1 \\[5pt] a_2 & b_2 & c_2 \\[5pt] a_3 & b_3 & c_3 \end{array}\right]}

is given by (expanding along {\text{R}_1})

{\left|\text{A}\right|}

{= \left|\begin{array}{cc} a_1 & b_1 & c_1 \\[5pt] a_2 & b_2 & c_2 \\[5pt] a_3 & b_3 & c_3 \end{array}\right|}

{= a_1\left|\begin{array}{cc} b_2 & c_2 \\[5pt] b_3 & c_3 \end{array}\right| - b_1\left|\begin{array}{cc} a_2 & c_2 \\[5pt] a_3 & c_3 \end{array}\right| + c_1 \left|\begin{array}{cc} a_2 & b_2 \\[5pt] a_3 & b_3 \end{array}\right|}

For any square matrix A, the |A| satisfy following properties.

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|A′| = |A|, where |A′| = transpose of A.

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If we interchange any two rows (or columns), then sign of determinant changes.

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If any two rows or any two columns are identical or proportional, then value of determinant is zero.

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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.}

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Multiplying a determnant by k means multiply elements of only one row (or one column) by {k.}

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If {\text{A} = {[a_{ij}]}_{3 × 3},} then {\left|k.\text{A}\right| = k^3\left|\text{A}\right|}

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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.

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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.

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Area of a triangle with vertices {\left(x_1, y_1\right),} {\left(x_2, y_2\right)} and {\left(x_3, y_3\right)} is given by

{Δ = \dfrac12 \left|\begin{array}{cc} x_1 & y_1 & 1 \\[5pt] x_2 & y_2 & 1 \\[5pt] x_3 & y_3 & 1 \end{array}\right|}

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Minor of an element a_{ij} of the determinant of matrix A is the determinant obtained by deleting i^{th} row and j^{th} column and denoted by \text{M}_{ij}.

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Cofactor of a_{ij} of given by {\text{A}_{ij} = \left(-1\right)^{i + j} \text{M}_{ij}}

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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,

{\left|\text{A}\right| = a_{11} \text{A}_{11} + a_{12} \text{A}_{12} + a_{13} \text{A}_{13}}

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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, {a_{11} \text{A}_{21} + a_{12} \text{A}_{22} + a_{13} \text{A}_{23} = 0}

If {\text{A} = \left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\[5pt] a_{21} & a_{22} & a_{23} \\[5pt] a_{31} & a_{32} & a_{33} \end{array}\right],} then {adj \text{ A} = \left[\begin{array}{ccc} \text{A}_{11} & \text{A}_{21} & \text{A}_{31} \\[5pt] \text{A}_{12} & \text{A}_{22} & \text{A}_{32} \\[5pt] \text{A}_{13} & \text{A}_{23} & \text{A}_{33} \end{array}\right],} where {\text{A}_{ij}} is cofactor of {a_{ij}.}

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{\text{A }(adj\text{ A}) = (adj\text{ A})\text{ A} = |\text{A I}|,} where A is a square matrix of order {n.}

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A square matrix A is said to be singular or non-singular according as |A| = 0 and |A| ≠ 0.

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If AB = BA = I, where B is square matrix, then B is called inverse of A. Also {\text{A}^{-1} = \text{B}} or {\text{B}^{-1} = \text{A}} and hence {\left(\text{A}^{-1}\right)^{-1} = \text{A}.}

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A square matrix A has inverse if and only if A is non-singular.

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{\text{A}^{-1} = \dfrac{1}{\text{|A|}}(adj\text{ A})}

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If

{a_1x + b_1y + c_1z = d_1}

{a_2x + b_2y + c_2z = d_2}

{a_3x + b_3y + c_3z = d_3,}

{a_2x + b_2y + c_2z = d_2}

{a_3x + b_3y + c_3z = d_3,}

then these equations can be written as A X = B, where

{\text{A} = \left[\begin{array}{cc} a_1 & b_1 & c_1 \\[5pt] a_2 & b_2 & c_2 \\[5pt] a_3 & b_3 & c_3 \end{array}\right],} {\text{X} = \left[\begin{array}{c} x \\[5pt] y \\[5pt] z \end{array}\right]} and {\left[\begin{array}{c} d_1 \\[5pt] d_2 \\[5pt] d_3 \end{array}\right]}

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Unique solution of equation AX = B is given by {\text{X = A}^{-1}\text{ B},} where |A| ≠ 0.

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A system of equation is consistent or inconsistent according as its solution exists or not.

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For a square matrix A in matrix equation AX = B

(i)

|A| ≠ 0, there exists unique solution

(ii)

|A| = 0, and {(adj\text{ A})\text{ B}} ≠ 0, then there exists no solution

(iii)

|A| = 0, and {(adj\text{ A})\text{ B}} = 0, then system may or may not be consistent.