Find all values of \(k\) such that the rank of the matrix \(\begin{pmatrix}
k \amp -1 \amp 0\amp 0 \\0 \amp k \amp -1\amp 0\\0\amp 0 \amp k\amp -1\\-6 \amp 11\amp -6 \amp 1
\end{pmatrix}\) is 3.
Find all values of \(k\) such that the rank of the matrix \(\begin{bmatrix} k+3 \amp 1 \amp -2\\ 3 \amp -2 \amp k\\-k \amp -3 \amp 3
\end{bmatrix}\) is 3.
For the following system of equations write the solution in the form of \(X_0+S_h\) where \(X_0\) is a solution of non homogeneous system and \(S_h\) is the set of solutions of the corresponding homogeneous system
Find a polynomial of the form \(y = a_0 +a_1 x+a_2 x^2 +a_3 x^3\) which passes through the points \((-3, -2), (-1, 2), (1, 5), (2, 1)\text{.}\)
Find the values of \(a\) and \(b\) for which the following system is consistent. Also find the complete solution when \(a = b = 2\text{.}\)
\begin{equation*}
x+y-z+w = 1;\quad ax + y + z + w = b;\quad
3x + 2y + aw = 1 + a\text{.}
\end{equation*}
The following table lists the number of milligrams of vitamin A, vitamin B, vitamin C, and niacin contained in 1 g of four different foods. A dietitian wants to prepare a meal that provides 250 mg of vitamin A, 300 mg of vitamin B, 400 mg of vitamin C, and 70 mg of niacin. Determine how many grams of each food must be included, and describe any limitations on the quantities of each food that can be used
Find the reduced row-echelon form of \(A\) and hence, or otherwise, prove that the solution of the above system is \(x_1 = x_2 = \ldots = x_n\) , with \(x_n\) arbitrary.
For which rational numbers \(a\) does the following system have (i) no solutions (ii) exactly one solution (iii) infinitely many solutions?
\begin{equation*}
x + 2y-3z = 4;\quad
3x-y + 5z = 2;\quad
4x + y + (a^2-14)z = a + 2\text{.}
\end{equation*}
If \((\alpha_1 ,\ldots,\alpha_n)\) and \((\beta_1 ,\ldots,\beta_n)\) are solutions of a system of linear equations, prove that
