Prove the following identities:
\[\begin{vmatrix}y + z & z & y \\ z & z + x & x \\ y & x & x + y\end{vmatrix} = 4xyz\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
\[\begin{vmatrix}- a \left( b^2 + c^2 - a^2 \right) & 2 b^3 & 2 c^3 \\ 2 a^3 & - b \left( c^2 + a^2 - b^2 \right) & 2 c^3 \\ 2 a^3 & 2 b^3 & - c \left( a^2 + b^2 - c^2 \right)\end{vmatrix} = abc \left( a^2 + b^2 + c^2 \right)^3\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
\[\begin{vmatrix}1 + a & 1 & 1 \\ 1 & 1 + a & a \\ 1 & 1 & 1 + a\end{vmatrix} = a^3 + 3 a^2\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove the following identity:
\[\begin{vmatrix}2y & y - z - x & 2y \\ 2z & 2z & z - x - y \\ x - y - z & 2x & 2x\end{vmatrix} = \left( x + y + z \right)^3\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove the following identity:
\[\begin{vmatrix}a + x & y & z \\ x & a + y & z \\ x & y & a + z\end{vmatrix} = a^2 \left( a + x + y + z \right)\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove the following identity:
`|(a^3,2,a),(b^3,2,b),(c^3,2,c)| = 2(a-b) (b-c) (c-a) (a+b+c)`
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Without expanding, prove that
\[\begin{vmatrix}a & b & c \\ x & y & z \\ p & q & r\end{vmatrix} = \begin{vmatrix}x & y & z \\ p & q & r \\ a & b & c\end{vmatrix} = \begin{vmatrix}y & b & q \\ x & a & p \\ z & c & r\end{vmatrix}\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Show that
\[\begin{vmatrix}x + 1 & x + 2 & x + a \\ x + 2 & x + 3 & x + b \\ x + 3 & x + 4 & x + c\end{vmatrix} =\text{ 0 where a, b, c are in A . P .}\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
\[If \begin{vmatrix}p & b & c \\ a & q & c \\ a & b & r\end{vmatrix} = 0,\text{ find the value of }\frac{p}{p - a} + \frac{q}{q - b} + \frac{r}{r - c}, p \neq a, q \neq b, r \neq c .\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Show that x = 2 is a root of the equation
\[\begin{vmatrix}x & - 6 & - 1 \\ 2 & - 3x & x - 3 \\ - 3 & 2x & x + 2\end{vmatrix} = 0\] and solve it completely.
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}x + a & b & c \\ a & x + b & c \\ a & b & x + c\end{vmatrix} = 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}x + a & x & x \\ x & x + a & x \\ x & x & x + a\end{vmatrix} = 0, a \neq 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}3x - 8 & 3 & 3 \\ 3 & 3x - 8 & 3 \\ 3 & 3 & 3x - 8\end{vmatrix} = 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}1 & x & x^2 \\ 1 & a & a^2 \\ 1 & b & b^2\end{vmatrix} = 0, a \neq b\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}x + 1 & 3 & 5 \\ 2 & x + 2 & 5 \\ 2 & 3 & x + 4\end{vmatrix} = 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}1 & x & x^3 \\ 1 & b & b^3 \\ 1 & c & c^3\end{vmatrix} = 0, b \neq c\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}1 & 1 & x \\ p + 1 & p + 1 & p + x \\ 3 & x + 1 & x + 2\end{vmatrix} = 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Solve the following determinant equation:
\[\begin{vmatrix}15 - 2x & 11 - 3x & 7 - x \\ 11 & 17 & 14 \\ 10 & 16 & 13\end{vmatrix} = 0\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Let \[A = \begin{bmatrix}3 & 2 & 7 \\ 1 & 4 & 3 \\ - 2 & 5 & 8\end{bmatrix} .\] Find matrices X and Y such that X + Y = A, where X is a symmetric and Y is a skew-symmetric matrix
[3] Matrices
Chapter: [3] Matrices
Concept: undefined >> undefined
Define a symmetric matrix. Prove that for
\[A = \begin{bmatrix}2 & 4 \\ 5 & 6\end{bmatrix}\],
A +
AT is a symmetric matrix where
AT is the transpose of
A.
[3] Matrices
Chapter: [3] Matrices
Concept: undefined >> undefined
