By using properties of determinants, show that:
\(\begin{vmatrix}a^2+1&ab&ac\\ab&b^2+1&bc\\ca&cb&c^2+1\end{vmatrix}\)=1+a2+b2+c2
By using properties of determinants, show that:
\(\begin{vmatrix}a^2+1&ab&ac\\ab&b^2+1&bc\\ca&cb&c^2+1\end{vmatrix}\)=1+a2+b2+c2
Solution and Explanation
△=\(\begin{vmatrix}a^2+1&ab&ac\\ab&b^2+1&bc\\ca&cb&c^2+1\end{vmatrix}\)
Taking out common factors a, b, and c from R1, R2, and R3 respectively, we have:
△=abc\(\begin{vmatrix}a+\frac{1}{a}&b&c\\a&b+\frac{1}{b}&c\\a&b&c+\frac{1}{c}\end{vmatrix}\)
Applying R2 \(\to\) R2 − R1 and R3 \(\to\) R3 − R1, we have:
△=abc\(\begin{vmatrix}a+\frac{1}{a}&b&c\\-\frac{1}{a}&\frac{1}{b}&0\\-\frac{1}{a}&0&\frac{1}{c}\end{vmatrix}\)
Applying C1 \(\to\) a C1, C2 → b C2, and C3 → c C3, we have
△=abc.\(\frac{1}{abc}\)\(\begin{vmatrix}a^2+1&b^2&c^2\\-1&1&0\\-1&0&1\end{vmatrix}\)
=\(\begin{vmatrix}a^2+1&b^2&c^2\\-1&1&0\\-1&0&1\end{vmatrix}\)
Expanding along R3, we have:
△=-1\(\begin{vmatrix}b^2&c^2\\1&0\end{vmatrix}+\begin{vmatrix}a^2+1&b^2\\-1&1\end{vmatrix}\)
=-1(-c2)+(a2+1+b2)=1+a2+b2+c2
Hence, the given result is proved.
Top CBSE CLASS XII Mathematics Questions
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Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Top CBSE CLASS XII Determinants Questions
Evaluate the determinants in Exercises 1 and 2.
\(\begin{vmatrix}2&4\\-5&-1\end{vmatrix}\)
Evaluate the determinants in Exercises 1 and 2.
(i) \(\begin{vmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{vmatrix}\)(ii) \(\begin{vmatrix}x^2&-x+1&x-1\\& x+1&x+1\end{vmatrix}\)
Using properties of determinants,prove that:
\(\begin{vmatrix} x & x^2 & 1+px^3\\ y & y^2 & 1+py^3\\z&z^2&1+pz^3 \end{vmatrix}\)\(=(1+pxyz)(x-y)(y-z)(z-x)\)- Using properties of determinants,prove that:
\(\begin{vmatrix}1& 1+p& 1+p+q\\ 2& 3+2p& 4+3p+2q\\ 3& 6+3p& 10+6p+3q\end{vmatrix}=1\) Using properties of determinants,prove that:
\(\begin{vmatrix} 3a& -a+b & -a+c\\ -b+a & 3b & -b+c \\-c+a&-c+b&3c\end{vmatrix}\)\(=3(a+b+c)(ab+bc+ca)\)
Top CBSE CLASS XII Questions
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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