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The shape BCDE (in blue) is a rectangle whose centre is at A(-8.5, 4.2) (in black). It is given that OA || BC, |BC| = 7.6 and |BE| = 5.4. For the following, enter your answers in the form (x, y) for both vectors and points. For example, if your answer is (-1.23, 4.56), enter (-1.23, 4.56) in the answer box.


2.(a) (i) Use row operations to find the inverse of (ii) Using the inverse matrix found above, solve the equation AX = B where (b) Use properties of determinants to (i) Factorise A=\left[\begin{array}{ccc} 1 & -1 & 2 \\ 3 & 2 & 4 \\ 0 & 1 & -2 \end{array}\right] B=\left[\begin{array}{l} 3 \\ 6 \\ 1 \end{array}\right] \left|\begin{array}{ccc} z^{2} & y^{2} & x^{2} \\ z & y & x \\ 3 & 3 & 3 \end{array}\right| \left|\begin{array}{ccc} 1 & b c & b c(b+c) \\ 1 & c a & c a(c+a) \\ 1 & a b & a b(a+b) \end{array}\right|=a b c\left|\begin{array}{lcc} a & 1 & b+c \\ b & 1 & c+a \\ c & 1 & a+b \end{array}\right| \begin{aligned} x+y-3 z+w &=& 2 \\ 2 x-y-3 z-2 w &=&-5 \\ x-3 y+z+4 w &=&-10 \\ 3 x+4 y-10 z-7 w &=& 9 \end{aligned} (ii) Prove, without expanding that (c) (i) Represent the following system of equations as an augmented matrix. (ii) Reduce the matrix in (c)(i) to row echelon form, find the rank and hence show that the system of equations admit an infinite number of solutions.Find a general solution for this system.


The matrix A=\left[\begin{array}{cc} -2 & -1 \\ 1 & 0 \end{array}\right] has one eigenvalue of multiplicity 2. Find this eigenvalue and the dimension of its associated eigenspace. The eigenvalue______ has an associated eigenspace with dimension __________.


\text { If } F(x, y)=\langle-2 \cdot x-y-1,-x+6 \cdot y-1\rangle \text {, then a potential function } f \text { such that } F=\nabla f \text { is: }


\text { The gradient vector field } \nabla f(x, y) \text { of the function } f(x, y)=2 \cdot x^{3} \cdot y-x \cdot y \text { is: }


(1 point) Find a function f and a number a such that 1+\int_{a}^{x} \frac{f(t)}{t^{7}} d t=2 x^{-3} f(x) =________ a =______


Question 5: (1 point) The vertices of a triangle ABC are given by position vectors a, b and c. where a=6i+4j+6k, b=3i+5j+6k, c=7i+6j+5k. Find the area of the triangle ABC giving your answer accurate to four significant figures.


Is w in the subspace spanned by (V₁, V₂, V₁]? Explain why? V1= [10-1], v2=[213], v3=[426] and w= [312]


Find eigenvalues and eigenvectors for the matrix \left[\begin{array}{cc} 16 & 5 \\ -30 & -9 \end{array}\right] The smaller eigenvalue _________has an eigenvector [ __ ___ ] The larger eigenvalue _____ has an eigenvector[ _____ ______ ]


3. [4 points] Let X be a vector field on Rm such that for every p = Rm the maximal integral curve of X going through p at time zero is defined on the whole of R. Consider the stereographic projection : Sm - {N} → Rm, where N denotes the North Pole. Let Y be a vector field on the sphere Sm such that its restriction to Sm{N} equals (0-¹),X. What can you say about YN, the value of Y at the North Pole?


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