Matrices

Matrices - Pandai Notes

MATHEMATICS • Form 5 • Chapter 2: Matrices

2.1 Matrices

A matrix is a powerful mathematical tool used to organise, display, and manipulate data systematically. In everyday situations, such as sales tracking, sports medals tally, and examination marks, data arranged in tables can be converted into rectangular or square arrays of numbers called matrices.

Learning Objectives

Represent information from real-world situations in the form of matrices.
Determine the order of a matrix.
Identify specific elements in a matrix using row and column notations (\(a_{ij}\)).
Determine whether two matrices are equal based on order and corresponding elements.

Anatomy of a Matrix: Order and Elements

See how rows, columns, matrix order (\(m \times n\)), and element positions (\(a_{ij}\)) are identified.

Matrix A Order: \(2 \times 3\)

Column 1

Column 2

Column 3

Row 1

\(a_{11}\)

\(a_{12}\)

\(a_{13}\)

Row 2

\(a_{21}\)

\(a_{22}\)

\(a_{23}\)

🔍 Element \(a_{23}\) is at the intersection of Row 2 and Column 3.

Brief Explanation

What is a Matrix?

A matrix is a set of numbers arranged in rows (horizontal) and columns (vertical) to form a rectangular or a square array. It is usually denoted by a capital letter and enclosed in brackets \([ \ ]\) or parentheses \(( \ )\).

Historical Fact

James Joseph Sylvester (1814–1897) first used the term "matrix" in 1850. Cuthbert Edmund Cullis (1875–1954) introduced brackets for matrices in 1913 and used the notation \(A = [a_{ij}]\).

Type of matrix	Description	Example order
Row matrix	Only one row	\(1 \times n\)
Column matrix	Only one column	\(m \times 1\)
Square matrix	Same number of rows and columns	\(2 \times 2\), \(3 \times 3\), \([6]\)
Rectangular matrix	Different number of rows and columns	\(2 \times 3\), \(3 \times 2\)

Count the rows

Count the horizontal lines of numbers. This gives \(m\).

Count the columns

Count the vertical lines of numbers. This gives \(n\).

Write the order

Write the order as \(m \times n\), read as "matrix \(m\) by \(n\)".

Identify elements

The element at row \(i\) and column \(j\) is denoted as \(a_{ij}\).

Try Answering First

Read the question, answer it in your mind, then press “Check Answer”.

State the order of matrix \(K = \begin{bmatrix} 12 & 9 & 1 \\ 5 & 10 & 7 \end{bmatrix}\).

Answer: Matrix \(K\) has 2 rows and 3 columns, so its order is \(2 \times 3\).

Given matrix \(B = \begin{bmatrix} 1 & -16 \\ 20 & 4 \end{bmatrix}\), find the value of \(b_{12} + b_{21}\).

Answer: \(b_{12} = -16\), \(b_{21} = 20\). Therefore, \(b_{12} + b_{21} = -16 + 20 = 4\).

Given that \(\begin{bmatrix} 6 & 0 \\ 3 & y \\ 2z - 3 & -5 \end{bmatrix} = \begin{bmatrix} x & 0 \\ 3 & 2 \\ -2 & -5 \end{bmatrix}\), determine the values of \(x\), \(y\), and \(z\).

Answer: \(x = 6\), \(y = 2\), \(2z - 3 = -2 \Rightarrow 2z = 1 \Rightarrow z = \frac{1}{2}\) (or \(0.5\)).

Common Mistakes

! Writing the matrix order as column × row instead of row × column.
! Reading \(a_{21}\) as “\(a\) twenty-one” instead of the individual indices “\(a\) two one” (row 2, column 1).
! Assuming two matrices are equal just because they contain the exact same numbers, even though their orders differ or the positions of the numbers are swapped.

Misconceptions

Misconception

A single-element matrix like \([6]\) is not a square matrix.

Correct Fact

\([6]\) is a square matrix because it has 1 row and 1 column (\(1 \times 1\)), so the number of rows equals the number of columns.

Misconception

Matrices with the same total number of elements are equal or interchangeable.

Correct Fact

Even if both matrices have 6 elements, a \(2 \times 3\) matrix and a \(3 \times 2\) matrix have different orders and cannot be equal.

Summary

A matrix is an arrangement of numbers in rows and columns enclosed in \([ \ ]\) or \(( \ )\).
Order of a matrix with \(m\) rows and \(n\) columns is written as \(m \times n\).
The element at row \(i\) and column \(j\) is denoted as \(a_{ij}\).
Two matrices are equal (\(A = B\)) if and only if both have the same order and all corresponding elements are equal.

Short Activity

Complete all parts first, then press the check answer button.

A. Objective Quiz

1 What is the order of the matrix \(\begin{bmatrix} 4 & -1 & 7 \\ 8 & 0 & 2 \\ 5 & 11 & 3 \end{bmatrix}\)?

A. \(3 \times 3\) B. \(3 \times 1\) C. \(1 \times 3\) D. \(2 \times 3\)

2 Given matrix \(P = \begin{bmatrix} 3 & -7 & 9 \end{bmatrix}\), what is the element \(p_{13}\)?

A. \(3\) B. \(-7\) C. \(9\) D. \(0\)

B. Fill in the Blanks

3 A matrix that has only one column is known as a matrix.

4 If \(\begin{bmatrix} 10 \\ 6y + 5 \end{bmatrix} = \begin{bmatrix} 5x \\ 2y - 9 \end{bmatrix}\), the value of \(x\) is .

C. Drag and Drop: Match the Matrix Form to Its Classification

Drag each matrix form to the correct classification. On a phone, tap the answer choice first, then tap the classification box.

Matrix Form / Example

\([15 \quad -8]\)

\(\begin{bmatrix} 6 \\ 9 \end{bmatrix}\)

\(\begin{bmatrix} 13 & 5 \\ 10 & 2 \end{bmatrix}\)

\(\begin{bmatrix} 76 & 82 & 72 \\ 80 & 88 & 70 \end{bmatrix}\)

Classification / Term

Row matrixCorrectTry againEmpty

Column matrixCorrectTry againEmpty

Square matrixCorrectTry againEmpty

Rectangular matrixCorrectTry againEmpty

Key Terms

Matrix Row Column Order of a matrix Element (\(a_{ij}\)) Row matrix Column matrix Square matrix Equal matrices

Matrices

Matrices - Pandai Notes

MATHEMATICS • Form 5 • Chapter 2: Matrices

2.1 Matrices

Learning Objectives

Represent information from real-world situations in the form of matrices.
Determine the order of a matrix.
Identify specific elements in a matrix using row and column notations (\(a_{ij}\)).
Determine whether two matrices are equal based on order and corresponding elements.

Anatomy of a Matrix: Order and Elements

See how rows, columns, matrix order (\(m \times n\)), and element positions (\(a_{ij}\)) are identified.

Matrix A Order: \(2 \times 3\)

Column 1

Column 2

Column 3

Row 1

\(a_{11}\)

\(a_{12}\)

\(a_{13}\)

Row 2

\(a_{21}\)

\(a_{22}\)

\(a_{23}\)

🔍 Element \(a_{23}\) is at the intersection of Row 2 and Column 3.

Brief Explanation

What is a Matrix?

Historical Fact

James Joseph Sylvester (1814–1897) first used the term "matrix" in 1850. Cuthbert Edmund Cullis (1875–1954) introduced brackets for matrices in 1913 and used the notation \(A = [a_{ij}]\).

Type of matrix	Description	Example order
Row matrix	Only one row	\(1 \times n\)
Column matrix	Only one column	\(m \times 1\)
Square matrix	Same number of rows and columns	\(2 \times 2\), \(3 \times 3\), \([6]\)
Rectangular matrix	Different number of rows and columns	\(2 \times 3\), \(3 \times 2\)

Count the rows

Count the horizontal lines of numbers. This gives \(m\).

Count the columns

Count the vertical lines of numbers. This gives \(n\).

Write the order

Write the order as \(m \times n\), read as "matrix \(m\) by \(n\)".

Identify elements

The element at row \(i\) and column \(j\) is denoted as \(a_{ij}\).

Try Answering First

Read the question, answer it in your mind, then press “Check Answer”.

State the order of matrix \(K = \begin{bmatrix} 12 & 9 & 1 \\ 5 & 10 & 7 \end{bmatrix}\).

Answer: Matrix \(K\) has 2 rows and 3 columns, so its order is \(2 \times 3\).

Given matrix \(B = \begin{bmatrix} 1 & -16 \\ 20 & 4 \end{bmatrix}\), find the value of \(b_{12} + b_{21}\).

Answer: \(b_{12} = -16\), \(b_{21} = 20\). Therefore, \(b_{12} + b_{21} = -16 + 20 = 4\).

Given that \(\begin{bmatrix} 6 & 0 \\ 3 & y \\ 2z - 3 & -5 \end{bmatrix} = \begin{bmatrix} x & 0 \\ 3 & 2 \\ -2 & -5 \end{bmatrix}\), determine the values of \(x\), \(y\), and \(z\).

Answer: \(x = 6\), \(y = 2\), \(2z - 3 = -2 \Rightarrow 2z = 1 \Rightarrow z = \frac{1}{2}\) (or \(0.5\)).

Common Mistakes

! Writing the matrix order as column × row instead of row × column.
! Reading \(a_{21}\) as “\(a\) twenty-one” instead of the individual indices “\(a\) two one” (row 2, column 1).
! Assuming two matrices are equal just because they contain the exact same numbers, even though their orders differ or the positions of the numbers are swapped.

Misconceptions

Misconception

A single-element matrix like \([6]\) is not a square matrix.

Correct Fact

\([6]\) is a square matrix because it has 1 row and 1 column (\(1 \times 1\)), so the number of rows equals the number of columns.

Misconception

Matrices with the same total number of elements are equal or interchangeable.

Correct Fact

Even if both matrices have 6 elements, a \(2 \times 3\) matrix and a \(3 \times 2\) matrix have different orders and cannot be equal.

Summary

A matrix is an arrangement of numbers in rows and columns enclosed in \([ \ ]\) or \(( \ )\).
Order of a matrix with \(m\) rows and \(n\) columns is written as \(m \times n\).
The element at row \(i\) and column \(j\) is denoted as \(a_{ij}\).
Two matrices are equal (\(A = B\)) if and only if both have the same order and all corresponding elements are equal.

Short Activity

Complete all parts first, then press the check answer button.

A. Objective Quiz

1 What is the order of the matrix \(\begin{bmatrix} 4 & -1 & 7 \\ 8 & 0 & 2 \\ 5 & 11 & 3 \end{bmatrix}\)?

A. \(3 \times 3\) B. \(3 \times 1\) C. \(1 \times 3\) D. \(2 \times 3\)

2 Given matrix \(P = \begin{bmatrix} 3 & -7 & 9 \end{bmatrix}\), what is the element \(p_{13}\)?

A. \(3\) B. \(-7\) C. \(9\) D. \(0\)

B. Fill in the Blanks

3 A matrix that has only one column is known as a matrix.

4 If \(\begin{bmatrix} 10 \\ 6y + 5 \end{bmatrix} = \begin{bmatrix} 5x \\ 2y - 9 \end{bmatrix}\), the value of \(x\) is .

C. Drag and Drop: Match the Matrix Form to Its Classification

Drag each matrix form to the correct classification. On a phone, tap the answer choice first, then tap the classification box.

Matrix Form / Example

\([15 \quad -8]\)

\(\begin{bmatrix} 6 \\ 9 \end{bmatrix}\)

\(\begin{bmatrix} 13 & 5 \\ 10 & 2 \end{bmatrix}\)

\(\begin{bmatrix} 76 & 82 & 72 \\ 80 & 88 & 70 \end{bmatrix}\)

Classification / Term

Row matrixCorrectTry againEmpty

Column matrixCorrectTry againEmpty

Square matrixCorrectTry againEmpty

Rectangular matrixCorrectTry againEmpty

Key Terms

Matrix Row Column Order of a matrix Element (\(a_{ij}\)) Row matrix Column matrix Square matrix Equal matrices

Matrices

2.1 Matrices

Learning Objectives

Anatomy of a Matrix: Order and Elements

Brief Explanation

What is a Matrix?

Historical Fact

Try Answering First

Common Mistakes

Misconceptions

Summary

Short Activity

A. Objective Quiz

B. Fill in the Blanks

C. Drag and Drop: Match the Matrix Form to Its Classification

Key Terms

Matrices

2.1 Matrices

Learning Objectives

Anatomy of a Matrix: Order and Elements

Brief Explanation

What is a Matrix?

Historical Fact

Try Answering First

Common Mistakes

Misconceptions

Summary

Short Activity

A. Objective Quiz

B. Fill in the Blanks

C. Drag and Drop: Match the Matrix Form to Its Classification

Key Terms

Chapter : Matrices

Topic : Matrices

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