, XII- MATHS Determinants - Results
Determinant.
To every square matrix A = [aij] of order n, we can associate a number (real or complex) called
determinant of the matrix A, written as det A, where aij is the (i, j)th element of A.
𝑎 𝑏 𝑎 𝑏
If 𝐴 = [ ], then determinant of A, denoted by |A| (or det A), given by |𝐴| = | | = ad – bc.
𝑐 𝑑 𝑐 𝑑
𝑎11 𝑎12 𝑎13
If 𝐴 = [𝑎21 𝑎22 𝑎23 ], then
𝑎31 𝑎32 𝑎33
𝑎11 𝑎12 𝑎13
|𝐴| = |𝑎21 𝑎22 𝑎23 | = 𝑎11 (𝑎22 . 𝑎33 − 𝑎22 . 𝑎23 ) − 𝑎12 (𝑎21 . 𝑎33 − 𝑎31 . 𝑎23 ) + 𝑎13 (𝑎21. 𝑎32 − 𝑎31 . 𝑎22 )
𝑎31 𝑎32 𝑎33
Determinant can be expanded along any row or column. The sign of the terms depends up on the
position in which the element lies. If the suffices add up to odd number, sign of the term is negative.
+ − +
For eg; sign of 𝑎32 is negative (3+2 =5, odd). |− + −|
+ − +
NOTE : Only square matrices have determinants.
Properties of Determinants
For any square matrix A, |A| satisfies the following properties.
(i) |A′| = |A|, where A′ is transpose of matrix A.
(ii) If we interchange any two rows (or columns), then sign of the determinant changes.
(iii) If any two rows or any two columns in a determinant are identical (or proportional), then
the value of the determinant is zero.
(iv) If we multiply each element of a row (or a column) of a determinant by constant k, then
value of the determinant is multiplied by k.
Multiplying a determinant by k means multiplying the elements of only
one row (or one column) by k.
If A is a matrix of order 3 x 3, then |𝒌 𝑨| = 𝒌𝟑 |𝑨|
If A is a matrix of order n x n, then |𝑘 𝐴| = 𝑘 𝑛 |𝐴|
(v) If elements of a row (or a column) in a determinant can be expressed as the sum of two or
more elements, then the given determinant can be expressed as the sum of two or more
determinants.
𝑎1 + 𝑘1 𝑏1 + 𝑘2 𝑐1 + 𝑘3 𝑎1 𝑏1 𝑐1 𝑘1 𝑘2 𝑘3
| 𝑎 2 𝑏 2 𝑐2 |=| 2𝑎 𝑏2 𝑐2 | + | 2 𝑏2
𝑎 𝑐2 |
𝑎3 𝑏3 𝑐3 𝑎3 𝑏3 𝑐3 𝑎3 𝑏3 𝑐3
(vi) If to each element of a row (or a column) of a determinant the equimultiples of
corresponding elements of other rows (columns) are added, then value of determinant
remains the same.
Points to remember while solving property based questions.
• While applying properties, try to get at least two zeroes in any row or column.
• Make it sure that at least 3 to 4 operations are done before expanding.
• Even if you got two zeroes in any row or column, do not expand without getting some terms
of the required answer.
• Try to get same terms on a row or column, so that you can take that common and make zero
easily.
© Dineshmaths 0509312916 XII Mathematics
Determinant.
To every square matrix A = [aij] of order n, we can associate a number (real or complex) called
determinant of the matrix A, written as det A, where aij is the (i, j)th element of A.
𝑎 𝑏 𝑎 𝑏
If 𝐴 = [ ], then determinant of A, denoted by |A| (or det A), given by |𝐴| = | | = ad – bc.
𝑐 𝑑 𝑐 𝑑
𝑎11 𝑎12 𝑎13
If 𝐴 = [𝑎21 𝑎22 𝑎23 ], then
𝑎31 𝑎32 𝑎33
𝑎11 𝑎12 𝑎13
|𝐴| = |𝑎21 𝑎22 𝑎23 | = 𝑎11 (𝑎22 . 𝑎33 − 𝑎22 . 𝑎23 ) − 𝑎12 (𝑎21 . 𝑎33 − 𝑎31 . 𝑎23 ) + 𝑎13 (𝑎21. 𝑎32 − 𝑎31 . 𝑎22 )
𝑎31 𝑎32 𝑎33
Determinant can be expanded along any row or column. The sign of the terms depends up on the
position in which the element lies. If the suffices add up to odd number, sign of the term is negative.
+ − +
For eg; sign of 𝑎32 is negative (3+2 =5, odd). |− + −|
+ − +
NOTE : Only square matrices have determinants.
Properties of Determinants
For any square matrix A, |A| satisfies the following properties.
(i) |A′| = |A|, where A′ is transpose of matrix A.
(ii) If we interchange any two rows (or columns), then sign of the determinant changes.
(iii) If any two rows or any two columns in a determinant are identical (or proportional), then
the value of the determinant is zero.
(iv) If we multiply each element of a row (or a column) of a determinant by constant k, then
value of the determinant is multiplied by k.
Multiplying a determinant by k means multiplying the elements of only
one row (or one column) by k.
If A is a matrix of order 3 x 3, then |𝒌 𝑨| = 𝒌𝟑 |𝑨|
If A is a matrix of order n x n, then |𝑘 𝐴| = 𝑘 𝑛 |𝐴|
(v) If elements of a row (or a column) in a determinant can be expressed as the sum of two or
more elements, then the given determinant can be expressed as the sum of two or more
determinants.
𝑎1 + 𝑘1 𝑏1 + 𝑘2 𝑐1 + 𝑘3 𝑎1 𝑏1 𝑐1 𝑘1 𝑘2 𝑘3
| 𝑎 2 𝑏 2 𝑐2 |=| 2𝑎 𝑏2 𝑐2 | + | 2 𝑏2
𝑎 𝑐2 |
𝑎3 𝑏3 𝑐3 𝑎3 𝑏3 𝑐3 𝑎3 𝑏3 𝑐3
(vi) If to each element of a row (or a column) of a determinant the equimultiples of
corresponding elements of other rows (columns) are added, then value of determinant
remains the same.
Points to remember while solving property based questions.
• While applying properties, try to get at least two zeroes in any row or column.
• Make it sure that at least 3 to 4 operations are done before expanding.
• Even if you got two zeroes in any row or column, do not expand without getting some terms
of the required answer.
• Try to get same terms on a row or column, so that you can take that common and make zero
easily.
© Dineshmaths 0509312916 XII Mathematics
