 # Determinant and Properties of Determinants

• Properties of determinants: If A is the square matrix of order n, then for any scalar k; |kA| = kn|A|.
• In the determinant, if the rows and columns are interchanged then the value of the determinant is unaltered i.e, |A| = |AT|
• The value of the determinant by using the cofactors of the second row is given by if A = \begin{vmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix}
⇒ |A| = a21 c21 + a22 c22 + a23 c23
• det (AB) = det (BA) = (det A) (det B)
• The sum of the product of the elements of any row (column) of a square matrix with the cofactors the corresponding elements of any other row (column) is zero.
\Delta = \begin{vmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix} \Rightarrow
a11 c21 + a12 c22 + a13 c23 = 0
a11 c31 + a12 c32 + a13 c33 = 0
a21 c11 + a22 c12 + a23 c13 = 0
a31 c11 + a32 c12 + a33 c13 = 0
a11 c31 + a12 c32 + a13 c33 = 0
a21 c31 + a22 c32 + a23 c33 = 0
• det (AB) = 0 ⇒ either det A = 0 or det B = 0
• The determinant of the triangular matrix is the product of the principal diagonal elements.
i.e. A = \begin{vmatrix}a_{1} & a_{2} & a_{3} \\ 0 & b_{2} & b_{3} \\ 0 & 0 & c_{3} \end{vmatrix} = a_{1}b_{2}c_{3}
A = \begin{vmatrix}a_{1} & 0 & 0 \\ b_{1} & b_{2} & 0 \\ c_{1} & c_{2} & c_{3} \end{vmatrix} = a_{1}b_{2}c_{3}
• The determinant of unit matrix is 1
• The determinant of skew symmetric matrix of order 3 is 0
i.e., A = \begin{vmatrix}0 & a_{1} & a_{3} \\ -a_{1} & 0 & c_{1} \\ -a_{3} & -c_{1} & 0 \end{vmatrix} = 0
• The determinant of a square matrix changes sign when any two rows (columns) are interchanged.
i.e. \begin{vmatrix}a_{1} & b_{1} \\ a_{2} & b_{2} \end{vmatrix} = - \begin{vmatrix}a_{2} & b_{2} \\ a_{1} & b_{1} \end{vmatrix}
• In the determinant, if any two two rows (columns) are identical (or) proportional, then the value of the determinant is 0.
i.e. \begin{vmatrix}a_{1} & b_{1} \\ 2a_{1} & 2b_{1} \end{vmatrix} = 0 {\tt (or)} \begin{vmatrix}a_{1} & b_{1} \\ a_{1} & b_{1} \end{vmatrix} = 0
• In the determinant, if all the elements of any row (column) are equal to 0, then the value of the determinant 0.
i.e. \begin{vmatrix}a_{1} & b_{1} & 0 \\ a_{2} & b_{2} & 0 \\ a_{3} & b_{3} & 0 \end{vmatrix} = 0
• If all the elements of a row (column) of a square matrix are multiplied by a number k, then the determinant of the resulting matrix is equal to k times the determinant of the original matrix. i.e.
\begin{vmatrix}ka_{1} & b_{1} & c_{1} \\ ka_{2} & b_{2} & c_{2} \\ ka_{3} & b_{3} & c_{3} \end{vmatrix} = k \begin{vmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix}
• If each element of row (column) of a square matrix is the sum of the two terms, then its determinant can be expressed as the sum of two determinants of two square matrices of the same order. i.e.
\begin{vmatrix}a_{1} + x & b_{1} & c_{1} \\ a_{2} + y & b_{2} & c_{2} \\ a_{3} + z & b_{3} & c_{3} \end{vmatrix} = \begin{vmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix} + \begin{vmatrix}x & b_{1} & c_{1} \\ y & b_{2} & c_{2} \\ z & b_{3} & c_{3} \end{vmatrix}
• If the elements of a row (column) of a square matrix are added with k times the corresponding elements of any other row (column), then the value of the determinant of the resulting matrix is unaltered. i.e.
\begin{vmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix} = \begin{vmatrix}x + ka_{1} & b_{1} & c_{1} \\ y + ka_{2} & b_{2} & c_{2} \\ z + ka_{3} & b_{3} & c_{3} \end{vmatrix}
• The area of the triangle formed by the vertices A(x1, y1) B(x2, y2) C(x3, y3) by using determinants is = \frac{1}{2} \begin{vmatrix}x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{vmatrix}
• The condition for the points A(x1, y1) B(x2, y2) and C(x3, y3) are collinear is \begin{vmatrix}x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{vmatrix} = 0
• The equation of a line joining two points A(x1, y1) B(x2, y2) by using determinants is \begin{vmatrix}x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{vmatrix} = 0
• A square matrix A is said to be singular matrix if |A| = 0
• A square matrix A is said to be nonsingular matrix if |A| ≠ 0
• Tips for determinant results
• \begin{vmatrix}a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = -(a^{3} + b^{3} + c^{3} - 3abc)
= -\frac{(a + b + c)}{2}\left[(a - b)^{2} + (b - c)^{2} + (c - a)^{2}\right]
• \begin{vmatrix}a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = abc + 2fgh - af^{2} - bg^{2} - ch^{2}
• \begin{vmatrix}1 & a & a^{2} \\ 1 & b & b^{2} \\ 1 & c & c^{2} \end{vmatrix} = \begin{vmatrix}1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{vmatrix} = (a - b) (b - c) (c - a)
• \begin{vmatrix}1 & a & a^{3} \\ 1 & b & b^{3} \\ 1 & c & c^{3} \end{vmatrix} = \begin{vmatrix}1 & 1 & 1 \\ a & b & c \\ a^{3} & b^{3} & c^{3} \end{vmatrix} = (a + b + c) (a - b) (b - c) (c - a)
• \begin{vmatrix}1 & a^{2} & a^{3} \\ 1 & b^{2} & b^{3} \\ 1 & c^{2} & c^{3} \end{vmatrix} = \begin{vmatrix}1 & 1 & 1 \\ a^{2} & b^{2} & c^{2} \\ a^{3} & b^{3} & c^{3} \end{vmatrix} = (ab + bc + ca) (a - b) (b - c) (c - a)
• \begin{vmatrix}a & b & c \\ a^{2} & b^{2} & c^{2} \\ a^{3} & b^{3} & c^{3} \end{vmatrix} = abc (a - b) (b - c) (c - a)
• If A = \begin{vmatrix}f(x) & g(x) & h(x) \\ A(x) & B(x) & C(x) \\ l(x) & m(x) & n(x) \end{vmatrix}
\frac{d}{dx} (A) = \begin{vmatrix}f'(x) & g'(x) & h'(x) \\ A(x) & B(x) & C(x) \\ l(x) & m(x) & n(x) \end{vmatrix} + \begin{vmatrix}f(x) & g(x) & h(x) \\ A'(x) & B'(x) & C'(x) \\ l(x) & m(x) & n(x) \end{vmatrix} + \begin{vmatrix}f(x) & g(x) & h(x) \\ A(x) & B(x) & C(x) \\ l'(x) & m'(x) & n'(x) \end{vmatrix}
• Product of the two determinants
• \begin{vmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix} \times \begin{vmatrix}A_{1} & B_{1} & C_{1} \\ A_{2} & B_{2} & C_{2} \\ A_{3} & B_{3} & C_{3} \end{vmatrix}
= \begin{vmatrix}a_{1}A_{1} + b_{1}B_{1} + c_{1}C_{1} & a_{1}A_{2} + b_{1}B_{2} + c_{1}C_{2} & a_{1}A_{3} + b_{1}B_{3} + c_{1}C_{3} \\ a_{2}A_{1} + b_{2}B_{1} + c_{2}C_{1} & a_{2}A_{2} + b_{2}B_{2} + c_{2}C_{2} & a_{2}A_{3} + b_{2}B_{3} + c_{2}C_{3} \\ a_{3}A_{1} + b_{3}B_{1} + c_{3}C_{1} & a_{3}A_{2} + b_{3}B_{2} + c_{3}C_{2} & a_{3}A_{3} + b_{3}B_{3} + c_{3}C_{3} \end{vmatrix}

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• If \tt A = \begin{bmatrix}a & b \\c & d \end{bmatrix} \Rightarrow |A| = ad - bc
• If \tt A = \begin{bmatrix}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{bmatrix}
⇒ |A| = a1 (b2 c3 − b3c2) − a2 (b1 c3 − b3c1) + a3 (b1 c2 − b2c1)