91Ó°ÊÓ

How many types of \(2 \times 3\) matrices in reduced rowechelon form are there? See Exercise 22

Consider the differential equation \\[ \frac{d^{2} x}{d t^{2}}-\frac{d x}{d t}-x=\cos (t) \\] This equation could describe a forced damped oscillator, as we will see in Chapter \(9 .\) We are told that the differential equation has a solution of the form \\[ x(t)=a \sin (t)+b \cos (t) \\] Find \(a\) and \(b,\) and graph the solution.

Let \(A\) be a \(4 \times 3\) matrix, and let \(\vec{b}\) and \(\vec{c}\) be two vectors in \(\mathbb{R}^{4}\). We are told that the system \(A \vec{x}=\vec{b}\) has a unique solution. What can you say about the number of solutions of the system \(A \vec{x}=\vec{c} ?\)

Is there a sequence of elementary row operations that transforms \\[ \left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right] \text { into }\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array}\right] ? \\] Explain.

Find a polynomial of degree \(\leq 2\) [of the form \(f(t)=\) \(\left.a+b t+c t^{2}\right]\) whose graph goes through the points \((1, p),(2, q),(3, r),\) where \(p, q, r\) are arbitrary constants. Does such a polynomial exist for all values of \(p, q, r ?\)

Find the polynomial of degree 3 [a polynomial of the form \(\left.f(t)=a+b t+c t^{2}+d t^{3}\right]\) whose graph goes through the points \((0,1),(1,0),(-1,0),\) and (2,-15) Sketch the graph of this cubic.

Find the polynomial of degree 4 whose graph goes through the points (1,1),(2,-1),(3,-59),(-1,5) and \((-2,-29) .\) Graph this polynomial.

Find the function \(f(t)\) of the form \(f(t)=a e^{3 t}+b e^{2 t}\) such that \(f(0)=1\) and \(f^{\prime}(0)=4\).

Consider the linear system \\[ \left|\begin{array}{l} x+y=1 \\ x+\frac{t}{2} y=t \end{array}\right| \\] where \(t\) is a nonzero constant. a. Determine the \(x-\) and \(y\) -intercepts of the lines \(x+y=1\) and \(x+(t / 2) y=t ;\) sketch these lines. For which values of the constant \(t\) do these lines intersect? For these values of \(t,\) the point of intersection \((x, y)\) depends on the choice of the constant \(t ;\) that is, we can consider \(x\) and \(y\) as functions of \(t .\) Draw rough sketches of these functions.Explain briefly how you found these graphs. Argue geometrically, without solving the system algebraically. b. Now solve the system algebraically. Verify that the graphs you sketched in part (a) are compatible with your algebraic solution.

Find the rank of the matrix $$\left[\begin{array}{lll} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{array}\right]$$ where \(a, d,\) and \(f\) are nonzero, and \(b, c,\) and \(e\) are arbitrary numbers.