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 a₁₁ is M₁₁ = \(\left|\begin{array}{ll}a_{22} & a_{23} \\a_{32} & a_{33}\end{array}\right|\), Similarly minor of a₁₂ is M₁₂ = \(\left|\begin{array}{ll}a_{21} & a_{23} \\a_{31} & a_{33}\end{array}\right|\)
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 F₁₁ = (-1)¹⁺¹ M₁₁ = M₁₁ = \(\left|\begin{array}{ll}a_{22} & a_{23} \\a_{32} & a_{33}\end{array}\right|\)
and F₁₂ = (-1)¹⁺² M₁₂ = – M₁₂ = –\(\left|\begin{array}{ll}a_{21} & a_{23} \\
a_{31} & a_{33}\end{array}\right|\)
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 = a₁₁F₁₁ + a₁₂F₁₂ + a₁₃F₁₃
• The sum of the product of element of any row with corresponding cofactor of another row is equal to zero i.e. a₁₁F₂₁ + a₁₂F₂₂ + a₁₃F₂₃ = 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) \(\frac{\mathrm{d}}{\mathrm{dx}}\left|\begin{array}{l}
\mathrm{R}_{1} \\\mathrm{R}_{2} \\\mathrm{R}_{3}\end{array}\right|=\left|\begin{array}{l}
\mathrm{R}_{1}^{\prime} \\\mathrm{R}_{2}\\\mathrm{R}_{3}\end{array}\right|+\left|\begin{array}{l}\mathrm{R}_{1} \\\mathrm{R}_{2}^{\prime} \\\mathrm{R}_{3}\end{array}\right|+\left|\begin{array}{l}\mathrm{R}_{1} \\\mathrm{R}_{2} \\
\mathrm{R}_{3}^{\prime}\end{array}\right|\)
R₁, R₂, R₃ represent row’s of determinant

(ii) \(\frac{\mathrm{d}}{\mathrm{dx}}\)|C₁ C₂ C₃| = |C’₁ C₂ C₃| + |C₁ C’₂ C₃| + |C₁ C₂ C’₃|
where C₁, C₂, C₃ 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’
a₁x + b₁y + c₁z = d₁ …….(i)
a₂x + b₂y + c₂z = d₂ …….(ii)
a₃x + b₃y + c₃z = d₃ …….(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 Δ₁, Δ₂, Δ₃ is not zero then the system of equations is inconsistent i.e. has no solution
(ii) if d₁ = d₂ = d₃ = 0 or Δ₁, Δ₂, Δ₃ are all zero then the system of equation is consistent and has infinitely many solutions.

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