If \[\begin{bmatrix}xy & 4 \\ z + 6 & x + y\end{bmatrix} = \begin{bmatrix}8 & w \\ 0 & 6\end{bmatrix}\] , write the value of (x + y + z).
[3] Matrices
Chapter: [3] Matrices
Concept: undefined >> undefined
If \[\binom{x + y}{x - y} = \begin{bmatrix}2 & 1 \\ 4 & 3\end{bmatrix}\binom{1}{ - 2}\] , then write the value of (x, y).
[3] Matrices
Chapter: [3] Matrices
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}1 & b + c & b^2 + c^2 \\ 1 & c + a & c^2 + a^2 \\ 1 & a + b & a^2 + b^2\end{vmatrix} = \left( a - b \right) \left( b - c \right) \left( c - a \right)\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}1 & a & bc \\ 1 & b & ca \\ 1 & c & ab\end{vmatrix} = \begin{vmatrix}1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2\end{vmatrix}\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}z & x & y \\ z^2 & x^2 & y^2 \\ z^4 & x^4 & y^4\end{vmatrix} = \begin{vmatrix}x & y & z \\ x^2 & y^2 & z^2 \\ x^4 & y^4 & z^4\end{vmatrix} = \begin{vmatrix}x^2 & y^2 & z^2 \\ x^4 & y^4 & z^4 \\ x & y & z\end{vmatrix} = xyz \left( x - y \right) \left( y - z \right) \left( z - x \right) \left( x + y + z \right) .\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}\left( b + c \right)^2 & a^2 & bc \\ \left( c + a \right)^2 & b^2 & ca \\ \left( a + b \right)^2 & c^2 & ab\end{vmatrix} = \left( a - b \right) \left( b - c \right) \left( c - a \right) \left( a + b + c \right) \left( a^2 + b^2 + c^2 \right)\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}\left( a + 1 \right) \left( a + 2 \right) & a + 2 & 1 \\ \left( a + 2 \right) \left( a + 3 \right) & a + 3 & 1 \\ \left( a + 3 \right) \left( a + 4 \right) & a + 4 & 1\end{vmatrix} = - 2\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}a^2 & a^2 - \left( b - c \right)^2 & bc \\ b^2 & b^2 - \left( c - a \right)^2 & ca \\ c^2 & c^2 - \left( a - b \right)^2 & ab\end{vmatrix} = \left( a - b \right) \left( b - c \right) \left( c - a \right) \left( a + b + c \right) \left( a^2 + b^2 + c^2 \right)\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
If \[I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}, J = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix} and B = \begin{bmatrix}\cos \theta & \sin \theta \\ - \sin \theta & \cos \theta\end{bmatrix}\] then B equals )
[3] Matrices
Chapter: [3] Matrices
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}1 & a^2 + bc & a^3 \\ 1 & b^2 + ca & b^3 \\ 1 & c^2 + ab & c^3\end{vmatrix} = - \left( a - b \right) \left( b - c \right) \left( c - a \right) \left( a^2 + b^2 + c^2 \right)\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}a^2 & bc & ac + c^2 \\ a^2 + ab & b^2 & ac \\ ab & b^2 + bc & c^2\end{vmatrix} = 4 a^2 b^2 c^2\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}x + 4 & x & x \\ x & x + 4 & x \\ x & x & x + 4\end{vmatrix} = 16 \left( 3x + 4 \right)\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}1 & 1 + p & 1 + p + q \\ 2 & 3 + 2p & 4 + 3p + 2q \\ 3 & 6 + 3p & 10 + 6p + 3q\end{vmatrix} = 1\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove that :
\[\begin{vmatrix}a & b - c & c - b \\ a - c & b & c - a \\ a - b & b - a & c\end{vmatrix} = \left( a + b - c \right) \left( b + c - a \right) \left( c + a - b \right)\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove that
\[\begin{vmatrix}a^2 & 2ab & b^2 \\ b^2 & a^2 & 2ab \\ 2ab & b^2 & a^2\end{vmatrix} = \left( a^3 + b^3 \right)^2\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
Prove that
\[\begin{vmatrix}a^2 + 1 & ab & ac \\ ab & b^2 + 1 & bc \\ ca & cb & c^2 + 1\end{vmatrix} = 1 + a^2 + b^2 + c^2\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
\[\begin{vmatrix}1 & a & a^2 \\ a^2 & 1 & a \\ a & a^2 & 1\end{vmatrix} = \left( a^3 - 1 \right)^2\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
\[\begin{vmatrix}a + b + c & - c & - b \\ - c & a + b + c & - a \\ - b & - a & a + b + c\end{vmatrix} = 2\left( a + b \right) \left( b + c \right) \left( c + a \right)\]
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
[4] Determinants
Chapter: [4] Determinants
Concept: undefined >> undefined
