Consider a matrix \(\rm A = \begin{bmatrix} 1 & 0 & 0 \\\ 0 & 4 & -2 \\\ 0 & 1 & 1 \end{bmatrix} \) The matrix A satisfies the equation 6A-1 = A2 + cA + dI, where c and d are scalars and I is the identity matrix. Then (c + d) is equal to
5
The characteristic polynomial is \((\lambda-1)(\lambda-2)(\lambda-3)=\lambda^3-6\lambda^2+11\lambda-6\). Cayley–Hamilton gives \(A^3-6A^2+11A-6I=0\). Multiplying by A⁻¹ yields \(6A^{-1}=A^2-6A+11I\). Thus c = −6,d = 11 and c+d = 5.
Important Questions from System of Linear Equations
Let $A=\begin{bmatrix} 1 & 1 & 1 \\ -1 & -1 & -1 \\ 0 & 1 & -1 \end{bmatrix}$ and $b = \begin{bmatrix} 1/3 \\ -1/3 \\ 0 \end{bmatrix}$, then the system of linear equations $Ax = b$ has
- Let $ \mathbf{A} = \begin{bmatrix} 1 & 1 \\ 1 & 3 \\ -2 & -3 \end{bmatrix} $ and $ \mathbf{b} = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} $. For $ \mathbf{Ax} = \mathbf{b} $ to be solvable, which one of the following options is the correct condition on $ b_1, b_2 $ and $ b_3 $:
The nine numbers x1, x2, x3 ... x9, are in ascending order. Their average m is strictly greater than all the first eight numbers. Which of the following is true?
For what value of k, the system linear equation has no solution
(3k + 1)x + 3y - 2 = 0
(k2 + 1)x + (k - 2)y - 5 = 0
A system of equations is said to be inconsistent if
