STUPIDSID
1 (a) (i)
Is the matrix \(\begin{bmatrix}1 &0 &4 &6 \\0 &1 &5 &7 \\0 &0 &0 &1 \end{bmatrix}\)in now echelon form or reduced row-echelon form?
2 M
1 (a) (ii)
Find the rank of the matrix \[ \begin{bmatrix}8 &0 &0 &16 \\0 &0 &0 &6 \\0 &9 &9 &9 \end{bmatrix} \]
2 M
1 (b) (i)
Find the inverse of the matrix\[ \begin{bmatrix}1&2 \\ 1 & 3 \end{bmatrix}\]
2 M
1 (b) (ii)
Find the Eigen values of the matrix AT. Is A an invertible? \[A= \begin{bmatrix}1 & 3 & 4 \\ 0& 0& 5 \\ 0&0&9 \end{bmatrix}\]
2 M
1 (c) (i)
Express the following quadratic forms in matrix notation.
i) x2-4xy+y2
ii) 2x2+6xy-5y2
i) x2-4xy+y2
ii) 2x2+6xy-5y2
3 M
1 (c) (ii)
If ϕ=3x2y-y3z2, find grad ϕ at the point (1, -2, -1).
3 M
2 (a) (i)
Find the length of the arc of the curve y=log sec x from \[ x=0 \ to \ x= \dfrac{\pi}{3} \]
3 M
2 (a) (ii)
Prove that \[ \iint_S \overrightarrow{F} \cdot \widehat{n}dS = 3v , \ if \ \overrightarrow{F} = x\widehat{i} + y \widehat{j}+ z \widehat{k} \] where S is any closed surface enclosing volume V.
3 M
2 (b)
\[ If \ \overrightarrow{F}= 3xy \widehat{i} - y^2 \widehat{j}; \ evaluate \ \int_c \overrightarrow{F}\cdot \overrightarrow{d}r. \] Where C is the arc of the Parabola y=2x2 from (0,0) to (1,2).
4 M
2 (c)
Show that \[ \overrightarrow{F} = (y^2 - z^2 + 3yz - 2x)\widehat{i} + (3xz + 2xy)\widehat{j}+ (3xy-2xz + 2z)\widehat{k} \] is both solenoidal and irrotational.
4 M
3 (a)
Solve the following homogeneous system of equations by using Gauss-Jorfan elimination
2x1+2x2-x3+x5=0,
-x1-x2+2x3-3x4+x5=0,
x1+x2-2x3-x5=0,
x3+x4+x5=0
2x1+2x2-x3+x5=0,
-x1-x2+2x3-3x4+x5=0,
x1+x2-2x3-x5=0,
x3+x4+x5=0
5 M
3 (b)
Use Cramer's rule to solve:-
x+5y=2
11x+y+2z=3
x+5y+2z=1
x+5y=2
11x+y+2z=3
x+5y+2z=1
5 M
3 (c)
What conditions must b1, b2, b3 satisfy in order for the system of equations to be consistent?
x1+2x+3x=b1
2x1+5x2+3x3=b2
x1+ 8x3=b3
x1+2x+3x=b1
2x1+5x2+3x3=b2
x1+ 8x3=b3
4 M
4 (a)
Show that the set of all 2×2 matrices of the form \[ \begin{bmatrix}a &1 \\1 &b \end{bmatrix} \] with addition defined by \[ \begin{bmatrix}a &1 \\ 1 &b \end{bmatrix} + \begin{bmatrix}c &1 \\ 1 &b \end{bmatrix} = \begin{bmatrix}a+c &1 \\ 1 &b+d \end{bmatrix} \] and scalar multiplication defined by \[ \begin{bmatrix}a&1 \\ 1 &b \end{bmatrix} = \begin{bmatrix}ka&1 \\ 1 &kb \end{bmatrix} \] is a vector space.
6 M
4 (b)
Determine which of the following are subspaces of R3.
(1) all the vector of the form (a,b,c) where b=a+c.
(2) all the vector of the form (a,b,c) where b=a+c+1.
(1) all the vector of the form (a,b,c) where b=a+c.
(2) all the vector of the form (a,b,c) where b=a+c+1.
4 M
4 (c)
Let v1=1-3x+2x2, v2=1-x+4x2, v3=1-7x. Show that the set S= {v1, v2, v3} is a basis for P2.
4 M
5 (a)
Use the inner product \[ (f,g)= \int^1_0 f(x) g(x) dx; \] to compute d(f,g) and ||f||. Where f(x) = x, g(x)=ex in C[0,1].
4 M
5 (b)
Let R2 have the Euclidean inner product. Use the Gram Schmidt process to transform the basis {u1, u2, u3} into an orthonormal basis u1=(1,1,1), u2=(-1,1,0), u3=(1,2,1)
5 M
5 (c)
Verify that the vectors u1=(1,-1,2,-1) u2=(-2,2,3,2), u3=(1,2,0,-1), u4=(1,0,0,1) form an Orthogonal basis for R4 with the Euclidean inner product. Also convert it to an orthonormal set by normalizing the vectors.
5 M
6 (a)
Find the eigen values of the eigenspaces of the matrix \[ \begin{bmatrix}0 &0 &-2 \\1 &2 &1 \\1 &0 &3 \end{bmatrix} \]
5 M
6 (b)
Find matrix P that diagonalizes \[ A = \begin{bmatrix}1&0 \\ 6 & -1 \end{bmatrix} . \] Also determine P-1 AP.
5 M
6 (c)
Using Cayley Hamilton theorem, find A-1; \[A= \begin{bmatrix}1 & 4 \\ 2&3\end{bmatrix} \]
4 M
7 (a)
\[ Let \ T_A: R^6 \to R^4 \ be \ multiplication \ by \\ A= \begin{bmatrix}-1 &2 &0 &4 &5 &-3 \\3 &-7 &2 &0 &1 &4 \\2
&-5 &2 &4 &6 &1 \\4 &-9 &2 &-4 &-4 &7 \end{bmatrix} \] Find the rank and nullity of TA
5 M
7 (b)
Consider the basis S={v1, v2} for R2, where v1=(-2,1) and v2=(-1,3). Let T:R2→R3, be the linear operator such that T(v1)= (-1, 2, 0) and T(v2)=(0, -3, 5) find the formula T(x1, x2), and use that formula to find T(2, -3).
5 M
7 (c)
Show that the following functions are linear transformations.
i) T:R2 → R2, where T(x,y)=(x+3y, 3x-y)
ii) T:R2 → R2, where T(x,y)=(2x-y, x-y)
i) T:R2 → R2, where T(x,y)=(x+3y, 3x-y)
ii) T:R2 → R2, where T(x,y)=(2x-y, x-y)
4 M
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