Determinants
* Let A be the set of square matrices and B is a set of real or complex numbers. If a function, ƒ: A → B is defined by ƒ(x) = b, such that X ∈ A and b ∈ B, then ƒ(x) is
called the determinant of X.
* The determinant of A is denoted by | A | , det (A) or ∆.
* If any row (or column) contain maximum number of zeros, then finding the determinant along the row (or columns) makes the calculation easier.
* If all the elements in a row (or a column) are zeros, then the determinant of that
matrix is zero.
* If M and N are two square matrices such that M = kN, where is a real number, then | M | = k^{n} | N | ,
where n is the order of the matrices M and N.
* At the time of expansion, a_{ij} can also be multiplied by + 1 or – 1, depending upon whether (I + j) is even or odd,
instead of multiplying it by (-1)^{i + j}.
Properties of Determinants – I
* The value of a determinant remains unchanged if its rows and columns are interchanged.
* The sign of the determinant of square matrix changes if any two rows (or columns) in the matrix are interchanged.
* If any two rows (or columns) of square matrix are identical, then the value of its determinant is zero.
Properties of Determinants – II
* If all the elements in a row (or a column) of square matrix are multiplied by a number, k, then the value of the determinant of the matrix obtained is k times the
determinant of given matrix.
* If all the elements in a row (or a column) of the determinant of the matrix are expressed as the sum of two or more terms, then the determinant can be expressed
as the sum of two or more determinants.
* If all the elements in a row (or a column) of square matrix are added with k times (where k is constant) the corresponding elements of another row (or a column),
then the value of the determinant of the given matrix does not change.
Area of a Triangle
* The area of a triangle formed by the vertices ( x_{1} , y_{1} ), ( x_{2} , y_{2} ) and ( x_{3} , y_{3} ) is
given by
\frac{1}{2} \begin{pmatrix}
x_{1} & y_{1} & {1}\\
x_{2} & y_{2} & {1}\\
x_{3} & y_{3} & {1}
\end{pmatrix} sq. units.
* While calculating area of a triangle by using the determinants formula, we need to take the absolute value of the determinant to avoid negative values, if any.
* If the area is given, then use both the positive and negative values of the determinant to calculate an unknown coordinate.
* The area of a triangle from by three collinear points is equal to zero.
Minors and Cofactors
* The minor of an element, a^{i + j} , of a matrix, A, can be obtained by finding the determinant of the matrix obtained by deleting the i^{th} row
and the j^{th} column. It is denoted by M_{ij}.
* The minor of an element of determinant of order n (n ≥ 2) is a determinant of (n – 1).
* Let a_{ij} be an element in the i^{th} row and the j^{th} column of a square matrix, A.
Cofactor of
a_{ij} = ( - 1)^{i + j}× M_{ij}
* The determinant of the matrix is equal to the sum of the products of the elements of a row or a column with their corresponding Cofactors.
* If the elements of a row (or column) are multiplied by the cofactors of any other row (or column), then their sum is zero.
Adjoint and inverse of a matrix
* The transpose of the matrix obtained by replacing the elements of a square matrix, A, by the corresponding Cofactors is called adjoint of a matrix A.
* If A is any given square matrix of order n, then A (adj A) = (adj A)A = | A | I,
where is the identity matrix of order n.
* A square matrix is said to be singular if its determinant is zero.
* A square matrix is said to be non-singular if its determinant is not zero.
* If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.
* The determinant of the product of matrices is equal to product of their respective
determinants, that is, | AB | = | A | | B | , where A and B are square matrices of the
same order.
* If A is a square matrices of order n, then | (adj)A | = | A | ^{n - 1}
* A square matrix, A, is said to be an invertible matrix if there exists a square matrix,
B, such that.
* If A is a non-singular matrix, then A is invertible and A ^{n - 1} = \frac{adj A}{det A}
Application of Determinants and Matrices
* The method of solving a system of linear equations by using the inverse of coefficient matrix is known as the matrix method.