Using the property of determinants and without expanding,prove that:\(\begin{vmatrix} b+c & q+r & y+z\\ c+a & r+p & z+x\\ a+b&p+q&x+y \end{vmatrix}\)=2\(\begin{vmatrix} a & p & x\\ b & q & y\\c&r&z \end{vmatrix}\)
Solution and Explanation
△=\(\begin{vmatrix} b+c & q+r & y+z\\ c+a & r+p & z+x\\ a+b&p+q&x+y \end{vmatrix}\)
=\(\begin{vmatrix} b+c & q+r & y+z\\ c+a & r+p & z+x\\ a&p&x \end{vmatrix}\)+\(\begin{vmatrix} b+c & q+r & y+z\\ c+a & r+p & z+x\\ b&q&y \end{vmatrix}\)
=△1+△2(say) ....(1)
Now △1= \(\begin{vmatrix} b+c & q+r & y+z\\ c+a & r+p & z+x\\ a&p&x \end{vmatrix}\)
Applying R2 → R2 − R3, we have:
△1=\(\begin{vmatrix} b+c & q+r & y+z\\ c & r & z\\ a&p&x \end{vmatrix}\)
Applying R1 → R1 − R2, we have:
△1=\(\begin{vmatrix} b & q & y\\ c & r & z\\ a&p&x \end{vmatrix}\)
Applying R1 ↔R3 and R2 ↔R3, we have:
△1=(-1)2\(\begin{vmatrix}a&p&x\\b&q&y\\c&r&z\end{vmatrix}=\begin{vmatrix}a&p&x\\b&q&y\\c&r&z\end{vmatrix}\) ….....(2)
△2=\(\begin{vmatrix} b+c & q+r & y+z\\ c+a & r+p & z+x\\ b&q&y \end{vmatrix}\)
Applying R1 → R1 − R3, we have:
△2=\(\begin{vmatrix} c & r & z\\ c+a & r+p & z+x\\ b&q&y \end{vmatrix}\)
Applying R2 → R2 − R1, we have:
△2=\(\begin{vmatrix} c & r & z\\ a & p & x\\ b&q&y \end{vmatrix}\)
Applying R1 ↔R2 and R2 ↔R3, we have:
△2=(-1)2\(\begin{vmatrix} a & p& x\\ b & q & y\\ c&r&z \end{vmatrix}\)= \(\begin{vmatrix} a & p& x\\ b & q & y\\ c&r&z \end{vmatrix}\) ...(3)
From (1), (2), and (3), we have:
△=2\(\begin{vmatrix} a & p& x\\ b & q & y\\ c&r&z \end{vmatrix}\) Hence, the given result is proved.
Top Questions on Determinants
Let I be the identity matrix of order 3 × 3 and for the matrix $ A = \begin{pmatrix} \lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2 \end{pmatrix} $, $ |A| = -1 $. Let B be the inverse of the matrix $ \text{adj}(A \cdot \text{adj}(A^2)) $. Then $ |(\lambda B + I)| $ is equal to _______
- JEE Main - 2025
- Mathematics
- Determinants
Let $A = \{ z \in \mathbb{C} : |z - 2 - i| = 3 \}$, $B = \{ z \in \mathbb{C} : \text{Re}(z - iz) = 2 \}$, and $S = A \cap B$. Then $\sum_{z \in S} |z|^2$ is equal to
- JEE Main - 2025
- Mathematics
- Determinants
If $ y(x) = \begin{vmatrix} \sin x & \cos x & \sin x + \cos x + 1 \\27 & 28 & 27 \\1 & 1 & 1 \end{vmatrix} $, $ x \in \mathbb{R} $, then $ \frac{d^2y}{dx^2} + y $ is equal to
- JEE Main - 2025
- Mathematics
- Determinants
- The number of singular matrices of order 2, whose elements are from the set $ \{2, 3, 6, 9\} $ is:
- JEE Main - 2025
- Mathematics
- Determinants
- Let the system of equations be: $ 2x + 3y + 5z = 9, $ $ 7x + 3y - 2z = 8, $ $ 12x + 3y - (4 + \lambda)z = 16 - \mu, $ which has infinitely many solutions. Then the radius of the circle centered at $ (\lambda, \mu) $ and touching the line $ 4x = 3y $ is:
- JEE Main - 2025
- Mathematics
- Determinants
Questions Asked in CBSE CLASS XII exam
- A 1 cm segment of a wire lying along the x-axis carries a current of 0.5 A along the \( +x \)-direction. A magnetic field \( \mathbf{B} = (0.4 \, \text{mT}) \hat{j} + (0.6 \, \text{mT}) \hat{k} \) is switched on in the region. The force acting on the segment is:
- CBSE CLASS XII - 2025
- Magnetic Effects of Current and Magnetism
The correct IUPAC name of \([ \text{Pt}(\text{NH}_3)_2\text{Cl}_2 ]^{2+} \) is:
- CBSE CLASS XII - 2025
- Coordination Compounds
- Using integration, find the area of the region bounded by the line \[ y = 5x + 2, \] the \( x \)-axis, and the ordinates \( x = -2 \) and \( x = 2 \).
- CBSE CLASS XII - 2025
- Evaluation of Definite Integrals by Substitution
- Two conductors A and B of the same material have their lengths in the ratio 1:2 and radii in the ratio 2:3. If they are connected in parallel across a battery, the ratio \( \frac{v_A}{v_B} \) of the drift velocities of electrons in them will be:
- CBSE CLASS XII - 2025
- Current electricity
- Arun, Bashir and Joseph were partners in a firm sharing profits and losses in the ratio of 5 : 3 : 2. They admitted Daksh as a new partner who acquired his share entirely from Arun. If Arun sacrificed \( \frac{1}{5} \)th from his share to Daksh, Daksh's share in the profits of the firm will be :
- CBSE CLASS XII - 2025
- Partnership Accounts
Concepts Used:
Properties of Determinants
Properties of Determinants:
- Reflection Property: The value of the determinant remains unchanged if its rows and columns are interchanged.
- Switching Property: The interchanging of any two rows (or columns) of the Determinant changes its signs.
- All- Zero Property: The Determinants will be equivalent to zero if each term of rows and columns is zero.
- Proportionality (Repetition) Property: If all elements of a row (or column) are proportional or identical to the elements of some other row (or column), then the determinant is zero.
- Scalar Multiple Property: If all the elements of a row (or column) of a determinant are multiplied by a non-zero constant, then the determinant gets multiplied by the same constant.
- Sum Property: If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.
- Property of Invariance: If to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same, i.e., the value of determinant remain same if we apply the operation Ri → Ri + kRj or Ci → Ci + kCj
- Factor Property: If a Determinant Δ becomes 0 while considering the value of x = α, then (x - α) is considered as a factor of Δ
- Triangle Property: If all the elements of a determinant above or below the main diagonal consist of zeros, then the determinant is equal to the product of diagonal elements.
Read More: Properties of Determinants