For ACT Students
The ACT is a timed exam...60 questions for 60 minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you
solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for any wrong answer.
For JAMB and CMAT Students
Calculators are not allowed. So, the questions are solved in a way that does not require a calculator.
Solve all questions.
Use at least two methods for each question as applicable.
Show all work.
(1.) CSEC $A,\:B\:and\:C$ are three $2$ x $2$ matrices such that
$
A = \begin{bmatrix}
a & b \\[3ex]
c & d
\end{bmatrix} \:\:
B = \begin{bmatrix}
5 & 3 \\[3ex]
3 & 2
\end{bmatrix} \:\:and\:\:
C = \begin{bmatrix}
14 & 0 \\[3ex]
-9 & 5
\end{bmatrix} \\[5ex]
$
Find
(i) $3A$
(ii) $B^{-1}$
(iii) $3A + B^{-1}$
(iv) the value of $a$, $b$, $c$, and $d$ given that $3A + B^{-1} = C$
$
(i)\:\: 3A = 3\begin{bmatrix}
a & b \\[3ex]
c & d
\end{bmatrix} =
\begin{bmatrix}
3(a) & 3(b) \\[3ex]
3(c) & 3(d)
\end{bmatrix} =
\begin{bmatrix}
3a & 3b \\[3ex]
3c & 3d
\end{bmatrix} \\[7ex]
(ii)\:\: B^{-1} \\[3ex]
$
Let us find the inverse in two ways.
Use any method you like.
These are the properties of reduced row echelon matrices.
Based on the properties:
Number (1.): The leading entry in any non-zero row is 1
Matrix D is out (7 in the second row).
Number (2.): All entries in the column above or below a leading 1 is zero
Matrices A (9 in the second column) and B (7 in the first column) are out.
Matrix C is the answer.
(3.) CSEC
(a) (i) (a)
Find the matrix product
$
\begin{bmatrix}
-1 & 3 \\[2ex]
4 & h
\end{bmatrix}
\begin{bmatrix}
k \\[2ex]
5
\end{bmatrix}
$
(b) Hence, find the values of $h$ and $k$ that satisfy the matrix equation.
(4.) Determine whether these statements are true or false.
The matrix $
\left[
\begin{array}{cc|c}
1 & 3 & -2 \\[3ex]
0 & 1 & 5 \\[3ex]
0 & 0 & 0
\end{array}
\right]$ is in row echelon form.
These are the properties of row echelon matrices.
A matrix is said to be in row echelon form (ref) or echelon form if each row has more leading zeros that the rows before it.
Based on this definition, the matrix is in row echelon form.
(5.) CSEC (a) Given that
$
D = \begin{bmatrix}
1 & 9p \\[3ex]
p & 4
\end{bmatrix}
$ is a singular matrix, determine the value(s) of $p$.
(b) Given the linear equations
$
2x + 5y = 6 \\[3ex]
3x + 4y = 8 \\[3ex]
$
(i) Write the equations in the form $AX = B$ where $A, X,\:\:and\:\:B$ are matrices.
(ii) (a) Calculate the determinant of the matrix $A$
(b) Show that
$
A^{-1} = \begin{bmatrix}
-\dfrac{4}{7} & \dfrac{5}{7} \\[5ex]
\dfrac{3}{7} & -\dfrac{2}{7}
\end{bmatrix} \\[3ex]
$
(c) Use the matrix $A^{-1}$ to solve for $x$ and $y$
A singular matrix is a matrix whose determinant is zero.
A. The system has no solution. B. The system has infinitely many solutions. C. The number of solutions cannot be determined. D. The system has one solution.
Equation (3.) from the augmented matrix can be written as:
0x + 0y + 0z = 4
This means that: 0 = 4
This is contradiction.
The system has no solution.
(7.)
(8.) Write the augmented matrix of the following systems of equations.
$
(a.)\;\;\begin{cases}
-8x + 7y + 4 = 0 \\[3ex]
-x + 9y - 1 = 0
\end{cases} \\[7ex]
(b.)\;\;\begin{cases}
0.07x - 0.09y = 0.04 \\[3ex]
0.18x + 0.30y = 0.4
\end{cases} \\[7ex]
(c.)\;\;\begin{cases}
x - y + z = 12 \\[3ex]
2x + 2y = 4 \\[3ex]
x + y + 9z = 3
\end{cases} \\[10ex]
(d.)\;\;\begin{cases}
x - y - z = -2 \\[3ex]
-7x + y - 3z = -1 \\[3ex]
2x + 8y = -5 \\[3ex]
3x + 9y + z = 0
\end{cases}
$
System of Equation: Variables on the LHS; constants on the RHS
Augmented Matrix: Coefficients of the variables on the LHS; constants on the RHS
(10.) (I.) Write the system of equations for the following augmented matrices.
(II.) Then, perform the indicated row operations on the augmented matrices.
You may verify your answers as applicable with the:
Matrices Calculators