Determinants Formula Sheet

A determinant is a scalar value that is computed from the elements of a square matrix and provides important properties about the matrix, such as whether it is invertible and information about the linear independence of its rows or columns.

Neetesh Kumar | April 28, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. What is a Minor?
• 2. What is a Cofactor?
• 3. Properties of Determinants
• 4. Multiplication of Two Determinants
• 5. Special Determinants
• 6. Solving System of Linear Equations

1. What is a Minor?:

The minor of a given element of a determinant is the determinant of the elements that remain after deleting the row & the column in which the given element stands.

For example, the minor of $a_1$ in $\begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}$ is $\begin{vmatrix} b_2 & c_2\\ b_3 & c_3\\ \end{vmatrix}$ & the minor of $b_2$ is $\begin{vmatrix} a_1 & c_1\\ a_3 & c_3\\ \end{vmatrix}$.
Hence, a determinant of order three will have "9 minors".

2. What is a Cofactor?:

If $M_{ij}$ represents the minor of the element belonging to $i^{th}$ row and $j^{th}$ column, then the cofactor of that element is calculated as:
$\bold{C_{ij} = (–1)^{i + j}. M_{ij}}$

Important Note:

Consider $\Delta$ = $\begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}$
Let $A_1$ be cofactor of $a_1$ and $B_2$ be cofactor of $b_2$ and so on, then
(i) $a_1A_1 + b_1B_1 + c_1C_1 = a_1A_1 + a_2A_2 + a_3A_3 = ............ = \Delta$
(ii) $a_2A_1 + b_2B_1 + c_2C_1 = b_1A_1 + b_2A_2 + b_3A_3 = ............. = 0$

3. Properties of Determinants :

$\bold{(a)}$ The value of a determinant remains unaltered if the rows & corresponding columns are interchanged.

$\bold{(b)}$ If any two rows (or columns) of a determinant are interchanged, the value of the determinant is changed only in the sign. For example
Let D = $\begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}$ & D' = $\begin{vmatrix} a_2 & b_2 & c_2\\ a_1 & b_1 & c_1\\ a_3 & b_3 & c_3 \end{vmatrix}$ Then D' = D

$\bold{(c)}$ If a determinant has two rows (or columns) identical or in the same proportion, its value is zero.

$\bold{(d)}$ If all the elements of any row (or column) are multiplied by the same number, then the determinant is multiplied by that number.

$\bold{(e)}$ $\begin{vmatrix} a_1 + x & b_1 + y & c_1 + z\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}$ = $\begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}$ + $\begin{vmatrix} x & y & z\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}$

$\bold{(f)}$ The value of a determinant is not altered by adding to the elements of any row (or column ) the same multiples of the corresponding elements of any other row (or column) For example
Let D = $\begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix}$ and D' = $\begin{vmatrix} a_1 + ma_2 & b_1 + mb_2 & c_1 + mc_2\\ a_2 & b_2 & c_2\\ a_3 + na_1 & b_3 + nb_1 & c_3 + nc_1 \end{vmatrix}$.
Then D' = D.

Note:

While applying this property $\bold{Atleast \space one \space ROW \space or \space COLUMN}$ must remain unchanged.

$\bold{(g)}$ If the elements of a determinant $\Delta$ are rational function of x and two rows (or columns) become identical when x = a, then (x - a) is a factor of $\Delta$.
Again, if r rows become identical when a is substituted for x, then $(x \space – \space a)^{r \space – \space 1}$ is a factor of $\Delta$.

$\bold{(h)}$ If D(x) = $\begin{vmatrix} f_1 & f_1 & f_1\\ g_2 & g_2 & g_2\\ h_3 & h_3 & h_3 \end{vmatrix}$, where $f_r$, $g_r$, $h_r;$ r = 1, 2, 3 are three differentiable functions. then
$\frac{d}{dx}$D(x) = $\begin{vmatrix} f'_1 & f'_1 & f'_1\\ g_2 & g_2 & g_2\\ h_3 & h_3 & h_3 \end{vmatrix}$ + $\begin{vmatrix} f_1 & f_1 & f_1\\ g'_2 & g'_2 & g'_2\\ h_3 & h_3 & h_3 \end{vmatrix}$ + $\begin{vmatrix} f_1 & f_1 & f_1\\ g_2 & g_2 & g_2\\ h'_3 & h'_3 & h'_3 \end{vmatrix}$

4. Multiplication of two Determinants:

$\begin{vmatrix} a_1 & b_1\\ a_2 & b_2\\ \end{vmatrix}$ X $\begin{vmatrix} l_1 & m_1\\ l_2 & m_2\\ \end{vmatrix}$ = $\begin{vmatrix} a_1l_1 + b_1l_2 & a_1m_1 + b_1m_2\\ a_2l_1 + b_2l_2 & a_2m_1 + b_2m_2\\ \end{vmatrix}$

Similarly, two determinants of order three can be multiplied.
$\bold{(a)}$ Here we have multiplied row by column. We can also multiply row by row, column by row, and column by column.

$\bold{(b)}$ If D' is the determinant formed by replacing the elements of determinant D of order n by their corresponding cofactors, then D' = D$^{n–1}$

5. Special Determinants:

(a) Symmetric Determinant:

Elements of a determinant are such that $a_{ij}$ = $a_{ji}$. For example $\begin{vmatrix} a & h & g\\ h & b & f\\ g & f & c \end{vmatrix}$ = abc + 2fgh - af$^2$ - bg$^2$ - ch$^2$

(b) Skew Symmetric Determinant:

If $a_{ij}$ = $-a_{ji}$, then the determinant is said to be a skew-symmetric determinant. Here, all the principal diagonal elements are zero.
The value of a skew-symmetric determinant of odd order is zero, and of even order is a perfect square. For example $\begin{vmatrix} 0 & b & -c\\ -b & 0 & a\\ c & -a & 0 \end{vmatrix} = 0$

(c) Other Important Determinants:

(i) $\begin{vmatrix} 1 & 1 & 1\\ a & b & c\\ bc & ac & ab \end{vmatrix}$ = $\begin{vmatrix} 1 & 1 & 1\\ a & b & c\\ a^2 & b^2 & c^2 \end{vmatrix}$ = (a-b)(b-c)(c-a)

(ii) $\begin{vmatrix} a & b & c\\ b & c & a\\ c & a & b \end{vmatrix}$ = $-$(a$^3$ + b$^3$ + c$^3$ $-$ 3abc)

6. Solving system of Linear equations :

(a) Nature of Solution for a System of equation involving two variables:

$\begin{rcases} a_1x + b_1y + c_1 = 0 \\ a_2x + b_2y + c_2 = 0 \end{rcases}⇒…$

$\bold{Case:1}$
$\bold{Consistent}$ (System of the equation has a Solution)
(a) $\bold{Unique \space Solution:}$ If $\bold{\frac{a_1}{a_2} \ne \frac{b_1}{b_2}}$ or $\bold{a_1b_2 - a_2b_1 \ne 0}$ (Equations representing $\bold{Intersecting \space Lines)}$
(b) $\bold{Infinite \space Solution:}$ If $\bold{\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}}$ (Equations representing $\bold{Coincident \space Lines)}$

$\bold{Case:2}$
$\bold{Inconsistent}$ (System of the equation has no Solution)
If $\bold{\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}}$ (Equations representing $\bold{Parallel \space disjoint \space Lines)}$

If $\Delta_1$ = $\begin{vmatrix} b_1 & c_1\\ b_2 & c_2\\ \end{vmatrix}$, $\Delta_2$ = $\begin{vmatrix} c_1 & a_1\\ a_2 & b_2\\ \end{vmatrix}$, then x = $\frac{\Delta_1}{\Delta}$, y = $\frac{\Delta_2}{\Delta}$

(b) Nature of Solution for a System of equations involving three variables :

$\begin{rcases} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \end{rcases}⇒…$
To solve this system, we first define the following determinants.
$\Delta$ = $\begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix},$ $\Delta_1$ = $\begin{vmatrix} d_1 & b_1 & c_1\\ d_2 & b_2 & c_2\\ d_3 & b_3 & c_3 \end{vmatrix},$ $\Delta_2$ = $\begin{vmatrix} a_1 & d_1 & c_1\\ a_2 & d_2 & c_2\\ a_3 & d_3 & c_3 \end{vmatrix},$ $\Delta_3$ = $\begin{vmatrix} a_1 & b_1 & d_1\\ a_2 & b_2 & d_2\\ a_3 & b_3 & d_3 \end{vmatrix},$

Now, the following algorithm is used to solve the system. We need to check the value of $\Delta$
$\bold{Case:1}$
If $\bold{\Delta \ne 0:}$ System is $\bold{Consistent}$ and has $\bold{Unique \space Solution}$ then, x = $\frac{\Delta_1}{\Delta}$, y = $\frac{\Delta_2}{\Delta}, z = \frac{\Delta_3}{\Delta}$

$\bold{Case:2}$
If $\bold{\Delta = 0:}$We need to check the values of $\Delta_1,$ $\Delta_2$ and $\Delta_3$
(a) If Atleast one of $\Delta_1,$ $\Delta_2$ and $\Delta_3$ is $\bold{not}$ Zero $\implies$ System is $\bold{Inconsistent}$.
(b) If $\Delta_1 =$ $\Delta_2 =$ $\Delta_3 = 0$ $\implies$ Put $\bold{z = t}$ and solve any two equations to get the values of x & y in terms of t
$\space \space \space \space$ If these values of x, y and z in terms of t satisfy third equation $\implies$ System is $\bold{Consistent}$ and has $\bold{Infinite \space number \space of \space Solutions}$.
$\space \space \space \space$ If The values of x, y, z doesn't satisfy third equation $\implies$ System is $\bold{Inconsistent}$.

Note:

(i) $\bold{Trivial \space Solution:}$ In the solution set of equations, if all the variables assume zero, then such a solution set is called a Trivial solution; otherwise, the solution is called a non-trivial solution.
(ii) If d$_1$ = d$_2$ = d$_3$ = 0, then the system of linear equations is known as the system of Homogeneous linear equations, which always possesses at least one solution (0, 0, 0).
(iii) If the system of homogeneous linear equation possesses a non-zero/nontrivial solution, then $\Delta$ = 0. In such cases, a given system has infinite solutions.

$\color{red} \bold{Related \space Pages:}$
Operation on Matrices
Matrices Formula Sheet