Unit notes and practice questions

Grade 11 Mathematics unit 4: Determinants and Their Properties

Everything here is for unit 4 of the Grade 11 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.

4
Cards: 10
Practice questions on unit 4: 207–250
The textbook pages for unit 4

What unit 4 covers

1.Determinants of Matrices of Order
2.Minors and Cofactors of Elements of Matrices
3.Properties of Determinants
4.Solutions of Systems of Linear Equations Using Cramer's Rule
5.Applications

Short notes for unit 4

The cards filed under this chapter on the Grade 11 Mathematics page. Open one at a time.

Area of a Triangle in xy-planeFormula

\text{Area} = \frac{1}{2} \left| \det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} \right|

\text{Area} = \frac{1}{2} \left| \det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} \right|

$\text{Area}$: Area of the triangle
$x_1, y_1$: Coordinates of first vertex
$x_2, y_2$: Coordinates of second vertex
$x_3, y_3$: Coordinates of third vertex

When you use it

Use this formula to find the area of a triangle given the coordinates of its three vertices.

Watch out

The determinant can evaluate to a negative value. Always take the absolute value to ensure the area is positive.

Drafted from Grade 11 Mathematics, pages 207-250, then checked twice before it went up

Collinear Points TestFormula

\det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} = 0

\det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} = 0

$x_1, y_1$: Coordinates of point A
$x_2, y_2$: Coordinates of point B
$x_3, y_3$: Coordinates of point C

When you use it

Use to test whether three given points lie on the exact same straight line.

Watch out

Do not confuse this with finding area. Here the determinant must strictly equal zero for collinearity.

Drafted from Grade 11 Mathematics, pages 207-250, then checked twice before it went up

Equation of Line Passing Through Two PointsFormula

\det \begin{pmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{pmatrix} = 0

\det \begin{pmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{pmatrix} = 0

$x, y$: Variables representing any point on the line
$x_1, y_1$: Coordinates of first known point
$x_2, y_2$: Coordinates of second known point

When you use it

Use to find the Cartesian equation of a line passing through two distinct given points.

Watch out

Keep x and y in the first row as variables. Do not substitute numerical values into them.

Drafted from Grade 11 Mathematics, pages 207-250, then checked twice before it went up

Matrix Inverse via AdjointFormula

A^{-1} = \frac{1}{\det(A)} \text{Adj}(A)

A^{-1} = \frac{1}{\det(A)} \text{Adj}(A)

$A^{-1}$: Inverse of square matrix A
$\det(A)$: Determinant of matrix A, provided it is not zero
$\text{Adj}(A)$: Adjoint matrix, transpose of the cofactor matrix

When you use it

Use to compute the inverse of an invertible matrix using cofactors and determinant.

Watch out

The adjoint matrix is the transpose of the cofactor matrix, so rows and columns must be swapped.

Drafted from Grade 11 Mathematics, pages 207-250, then checked twice before it went up

Practice questions on unit 4

10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.

Question 1

What is another name for an invertible matrix?

A.A singular matrix
B.A nonsingular matrix
C.A zero matrix
D.A row matrix

Show the answer and why

Answer: B A nonsingular matrix

Why

By definition, an invertible matrix is also called a nonsingular matrix, while a matrix that is not invertible is called a singular matrix.

Question 2

If A and B are invertible matrices of the same size, which formula correctly gives the inverse of their product, (AB)^-1?

A.A^-1 B^-1
B.B^-1 A^-1
C.(BA)^-1
D.A^t B^t

Show the answer and why

Answer: B B^-1 A^-1

Why

According to the properties of invertible matrices, the inverse of the product is reversed: (AB)^-1 = B^-1A^-1.

Question 3

Which term describes a diagonal matrix in which all main diagonal elements are equal?

A.Scalar matrix
B.Column matrix
C.Upper triangular matrix
D.Zero matrix

Show the answer and why

Answer: A Scalar matrix

Why

A diagonal matrix where all elements along the main diagonal are equal is called a scalar matrix.

Question 4

Which of the following is NOT one of the three standard elementary row operations?

A.Swapping two rows
B.Multiplying a row by a non-zero constant
C.Adding a constant multiple of one row to another row
D.Squaring all entries in a single row

Show the answer and why

Answer: D Squaring all entries in a single row

Why

The three elementary row operations are swapping rows, scaling a row by a non-zero constant, and pivoting (adding a multiple of one row to another). Squaring entries is not an elementary row operation.

Question 5

Given the matrix A = [[1, 1], [1, 2]], what is its inverse A^-1?

A.[[2, -1], [-1, 1]]
B.[[-2, 1], [1, -1]]
C.[[1, -1], [-1, 2]]
D.[[2, 1], [1, 1]]

Show the answer and why

Answer: A [[2, -1], [-1, 1]]

Why

Using elementary row operations to reduce (A|I₂), we find that A^-1 = [[2, -1], [-1, 1]].

Question 6

If the inverse of matrix A is A^-1 = [[1, 1], [1, 2]], what is the inverse of its transpose, (A^t)^-1?

A.[[2, -1], [-1, 1]]
B.[[1, 1], [1, 2]]
C.[[1, 2], [1, 1]]
D.[[2, 1], [1, 1]]

Show the answer and why

Answer: B [[1, 1], [1, 2]]

Why

By the properties of invertible matrices, (A^t)^-1 = (A^-1)^t. Transposing A^-1 = [[1, 1], [1, 2]] gives [[1, 1], [1, 2]].

Question 7

Why is the matrix B = [[1, 2], [2, 4]] not invertible?

A.It is not a square matrix.
B.Its entries are all positive numbers.
C.Row reduction of (B|I_2) produces a row of zeros in the left matrix, so it cannot reduce to I_2.
D.Its transpose is equal to itself.

Show the answer and why

Answer: C Row reduction of (B|I_2) produces a row of zeros in the left matrix, so it cannot reduce to I_2.

Why

Applying the row operation R₂ → R₂ - 2R₁ transforms the left matrix to have a bottom row of all zeros. Since it cannot be reduced to the identity matrix I₂, matrix B is singular (not invertible).

Question 8

A laboratory technician wants to mix a 30% salt solution with a 20% salt solution to obtain 500 liters of a 25% salt solution. How many liters of each solution are needed?

A.200 liters of 30% and 300 liters of 20%
B.250 liters of 30% and 250 liters of 20%
C.150 liters of 30% and 350 liters of 20%
D.100 liters of 30% and 400 liters of 20%

Show the answer and why

Answer: B 250 liters of 30% and 250 liters of 20%

Why

Setting up the system: x + y = 500 and 0.3x + 0.2y = 125. Solving this system gives y = 250 liters and x = 250 liters.

Question 9

Given the 3x3 matrix C = [[1, 1, 1], [0, 1, 2], [0, 0, 1]], what is its inverse C^-1?

A.[[1, -1, 1], [0, 1, -2], [0, 0, 1]]
B.[[1, 1, -1], [0, 1, 2], [0, 0, 1]]
C.[[1, 0, 0], [-1, 1, 0], [1, -2, 1]]
D.[[-1, 1, -1], [0, -1, 2], [0, 0, -1]]

Show the answer and why

Answer: A [[1, -1, 1], [0, 1, -2], [0, 0, 1]]

Why

By augmenting with I₃ and applying row operations (R₁ → R₁ - R₂ and R₂ → R₂ - 2R₃), we obtain C^-1 = [[1, -1, 1], [0, 1, -2], [0, 0, 1]].

Question 10

In a triangle, the smallest angle measures 10 degrees more than half of the largest angle. The middle angle measures 12 degrees more than the smallest angle. What are the measures of the three angles?

A.40 degrees, 52 degrees, and 88 degrees
B.47 degrees, 59 degrees, and 74 degrees
C.45 degrees, 57 degrees, and 78 degrees
D.50 degrees, 62 degrees, and 68 degrees

Show the answer and why

Answer: B 47 degrees, 59 degrees, and 74 degrees

Why

Let x, y, z be the angles. The equations are x + y + z = 180, x - 0.5z = 10, and -x + y = 12. Solving using row operations yields z = 74 degrees, y = 59 degrees, and x = 47 degrees.