# Determinants Formulas

Worried to simplify the determinants calculations? Not anymore, here is a quite simple way to solve the problems on determinants. Its nothing but using the Determinant Formulas to compute basic to complex problems. By remembering the Determinant formulas, you can instantly simplify the calculations and get familiar with the Determinant concepts. Go through the further sections and find the Determinant Formulae Tables, Sheets, etc.

## Cheatsheet & Tables of Determinants Formulas

Get the list of Determinants formulas from here and solve the problems easily to score more marks in your Board exams. Also, you can finish your homework and math assignments quickly by memorizing the Determinant Formulas using formula sheet & tables. So, make use of this Determinants Formulae List & master in solving the problems like Multiplication of two Determinants, Skew Symmetric Determinant, etc.

**1. Minor and Cofactor**

If Δ = \(\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\

a_{31} & a_{32} & a_{33}\end{array}\right|\) then Minor of a_{11} is M_{11} = \(\left|\begin{array}{ll}a_{22} & a_{23} \\a_{32} & a_{33}\end{array}\right|\), Similarly minor of a_{12} is M_{12} = \(\left|\begin{array}{ll}a_{21} & a_{23} \\a_{31} & a_{33}\end{array}\right|\)

The cofactor of an element a_{ij} is denoted by F_{ij} & is equal to (-1)^{i+j} M_{ij} where M is a Minor of element a_{ij}.

If Δ = \(\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\a_{21} & a_{22} & a_{23} \\a_{31} & a_{32} & a_{33}\end{array}\right|\)

then F_{11} = (-1)^{1+1} M_{11} = M_{11} = \(\left|\begin{array}{ll}a_{22} & a_{23} \\a_{32} & a_{33}\end{array}\right|\)

and F_{12} = (-1)^{1+2} M_{12} = – M_{12} = –\(\left|\begin{array}{ll}a_{21} & a_{23} \\

a_{31} & a_{33}\end{array}\right|\)

Note:

- The sum of products of the element of any row with their corresponding cofactor is equal to the value of determinant i.e. Δ = a
_{11}F_{11}+ a_{12}F_{12}+ a_{13}F_{13} - The sum of the product of element of any row with corresponding cofactor of another row is equal to zero i.e. a
_{11}F_{21}+ a_{12}F_{22}+ a_{13}F_{23}= 0 - If order of a determinant (Δ) is ‘n’ then the value of the determinant formed by replacing every element by its cofactor is Δ
^{n-1}.

**2. Multiplication of two Determinants**

Multiplication of two determinants is possible if they are of same order. Multiplication of two third order Determinants is defined as follows

\(\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\

a_{3} & b_{3} & c_{3}\end{array}\right| \times\left|\begin{array}{lll}\ell_{1} & m_{1} & n_{1} \\

\ell_{2} & m_{2} & n_{2} \\\ell_{3} & m_{3} & n_{3}\end{array}\right|\)

= \(\left|\begin{array}{lll}a_{1} \ell_{1}+b_{1} \ell_{2}+c_{1} \ell_{3} & a_{1} m_{1}+b_{1} m_{2}+c_{1} m_{3} & a_{1} n_{1}+b_{1} n_{2}+c_{1} n_{3} \\a_{2} \ell_{1}+b_{2} \ell_{2}+c_{2} \ell_{3} & a_{2} m_{1}+b_{2} m_{2}+c_{2} m_{3} & a_{2} n_{1}+b_{2} n_{2}+c_{2} n_{3} \\a_{3} \ell_{1}+b_{3} \ell_{2}+c_{3} \ell_{3} & a_{3} m_{1}+b_{3} m_{2}+c_{3} m_{3} & a_{3} n_{1}+b_{3} n_{2}+c_{3} n_{3}\end{array}\right|\)

**3. Symmetric Determinant**

A determinant is called symmetric determinant if for its every element a_{ij} = a_{ji} ∀ i, j

**4. Skew Symmetric Determinant**

A determinant is called skew symmetric determinant if for its every element a_{ji} = – a_{ji} ∀ i, j

**5. Properties of determinants:**

(i) The determinant remains unaltered if its rows and columns are interchanged.

(ii) The interchange of any two rows (columns) in Δ changes its sign.

(iii) If all the element of a row in A are zero or two rows (columns) are identical (or proportional), then the value of Δ is zero.

(iv) If all the elements of one row (or column) is multiplied by a non zero No K, then value of the new determinant is “K” times the value of the original determinant.

(v) If the elements of a row (column) of a determinant are multiplied by a non zero number K and then added to the corresponding elements of another row (column) then the value of the determinants remain unaltered.

(vi) If Δ becomes zero on putting n = α then we say that (n – α) is a factor of Δ.

(vii) \(\left|\begin{array}{lll}a_{1}+\lambda_{1} & b_{1} & c_{1} \\a_{2}+\lambda_{2} & b_{2} & c_{2} \\a_{3}+\lambda_{3} & b_{3} & c_{3}\end{array}\right|=\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{3} & c_{3}\end{array}\right|+\left|\begin{array}{lll}\lambda_{1} & b_{1} & c_{1} \\\lambda_{2} & b_{2} & c_{2} \\\lambda_{3} & b_{3} & c_{3}\end{array}\right|\)

(viii) \(\left|\begin{array}{ccc}a_{1} & a_{2} & a_{3} \\0 & b_{2} & b_{3} \\0 & 0 & c_{3}

\end{array}\right|=a_{1} b_{2} c_{3}=\left|\begin{array}{ccc}a_{1} & 0 & 0 \\b_{1} & b_{2} & 0 \\c_{1} & c_{2} & c_{3}\end{array}\right|=\left|\begin{array}{ccc}a_{1} & 0 & 0 \\0 & b_{2} & 0 \\0 & 0 & c_{3}\end{array}\right|\)

(ix) \(\left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \\a_{3} & b_{2} & c_{3}\end{array}\right| \times\left|\begin{array}{lll}\alpha_{1} & \beta_{1} & \gamma_{1} \\\alpha_{2} & \beta_{2} & \gamma_{2} \\\alpha_{3} & \beta_{2} & \gamma_{3}\end{array}\right|\)

= \(\left|\begin{array}{lll}a_{1} \alpha_{1}+b_{1} \beta_{1}+c_{1} \gamma_{1} & a_{1} \alpha_{2}+b_{1} \beta_{2}+c_{1} \gamma_{2} & a_{1} \alpha_{3}+b_{1} \beta_{3}+c_{1} \gamma_{3} \\a_{2} \alpha_{1}+b_{2} \beta_{1}+c_{2} \gamma_{1} & a_{2} \alpha_{2}+b_{2} \beta_{2}+c_{2} \gamma_{2} & a_{2} \alpha_{3}+b_{2} \beta_{3}+c_{2} \gamma_{3} \\a_{3} \alpha_{1}+b_{3} \beta_{1}+c_{3} \gamma_{1} & a_{3} \alpha_{2}+b_{3} \beta_{2}+c_{3} \gamma_{2} & a_{3} \alpha_{3}+b_{3} \beta_{3}+c_{3} \gamma_{3}\end{array}\right|\)

**6. Differentiation of a determinant: There are two way’s of differentiating a determinant**

(i) \(\frac{\mathrm{d}}{\mathrm{dx}}\left|\begin{array}{l}

\mathrm{R}_{1} \\\mathrm{R}_{2} \\\mathrm{R}_{3}\end{array}\right|=\left|\begin{array}{l}

\mathrm{R}_{1}^{\prime} \\\mathrm{R}_{2}\\\mathrm{R}_{3}\end{array}\right|+\left|\begin{array}{l}\mathrm{R}_{1} \\\mathrm{R}_{2}^{\prime} \\\mathrm{R}_{3}\end{array}\right|+\left|\begin{array}{l}\mathrm{R}_{1} \\\mathrm{R}_{2} \\

\mathrm{R}_{3}^{\prime}\end{array}\right|\)

R_{1}, R_{2}, R_{3} represent row’s of determinant

(ii) \(\frac{\mathrm{d}}{\mathrm{dx}}\)|C_{1} C_{2} C_{3}| = |C’_{1} C_{2} C_{3}| + |C_{1} C’_{2} C_{3}| + |C_{1} C_{2} C’_{3}|

where C_{1}, C_{2}, C_{3} represents column of determinant.

**7. Integration of Determinant:**

If Δ any single row or column contain elements which are functions of x, then integration can be done in determinant format only as following.

If Δ = \(\left|\begin{array}{ccc}f(x) & g(x) & h(x) \\a & b & c \\p & q & r\end{array}\right|\) a b c where a, b, c, p, q, r all are constants, then

\(\int_{a}^{b} \Delta d x=\left|\begin{array}{cccc}\int_{a}^{b} & f(x) d x & \int_{a}^{b} g(x) d x & \int_{a}^{b} h(x) d x \\a & b & c \\p & q & r\end{array}\right|\)

If elements are not like above it is advisable to simplify determinants then integrate it.

**8. Crammer’s Rule**

Consider three linear simultaneous equation in ‘x’, ‘y’, ‘z’

a_{1}x + b_{1}y + c_{1}z = d_{1} …….(i)

a_{2}x + b_{2}y + c_{2}z = d_{2} …….(ii)

a_{3}x + b_{3}y + c_{3}z = d_{3} …….(iii)

i.e. \(x=\frac{\Delta_{1}}{\Delta}, y=\frac{\Delta_{2}}{\Delta}, z=\frac{\Delta_{3}}{\Delta}\)

Case-I If Δ ≠ 0 then x = \(\frac{\Delta_{1}}{\Delta}, y=\frac{\Delta_{2}}{\Delta}, z=\frac{\Delta_{3}}{\Delta}\)

∴ The system is consistent and has unique solutions

Case-II If Δ = 0 and

(i) if at least one of Δ_{1}, Δ_{2}, Δ_{3} is not zero then the system of equations is inconsistent i.e. has no solution

(ii) if d_{1} = d_{2} = d_{3} = 0 or Δ_{1}, Δ_{2}, Δ_{3} are all zero then the system of equation is consistent and has infinitely many solutions.