A-Level Further Mathematics
Core Pure 1 · Edexcel 9FM0
Matrices
Revision & Exam Pack
Complete specification coverage • Worked examples
Common examiner traps • One-page cheat sheet
Designed for A-Level Further Mathematics students · Edexcel specification
,A-Level Further Maths | Core Pure 1: Matrices Edexcel 9FM0
Specification Map
This pack covers every Matrices learning objective in Edexcel A-Level Further Mathematics
Core Pure 1.
Spec Content Section
6.1 Matrices: addition, subtraction, scalar multiplication, multiplica- §1, §2
tion
6.2 Zero and identity matrices §1
6.3 Linear transformations in 2D §5
6.4 Successive transformations §5
6.5 Invariant points and invariant lines §5
6.6 Determinants (2×2 and 3×3); singular matrices §3
6.7 Inverse matrices (2×2 and 3×3) §4
6.8 Solving three linear simultaneous equations in three variables §6
6.9 Geometrical interpretation of solutions of three planes §6
Quick Tip
Work through this pack actively — pause at each Worked Example and try the problem before
reading the solution. The Examiner Trap boxes pinpoint the mistakes that cost marks in past
papers; learn to recognise them before exam day.
2
, A-Level Further Maths | Core Pure 1: Matrices Edexcel 9FM0
1. Matrix Fundamentals
A matrix is a rectangular array of numbers arranged in rows and columns. We write matrices using
bold capital letters: A, B, M.
The order (or dimensions) of a matrix with m rows and n columns is written m × n. The entry in
the ith row and jth column of A is denoted aij .
Matrix Notation
a11 a12 ··· a1n
a21
a22 ··· a2n
A=
.. .. .. ..
. . . .
am1 am2 · · · amn
A has order m × n. We say “m by n”.
1.1 Special matrices
• Square matrix: same number of rows and columns (n × n).
• Zero matrix 0: every entry is 0. Acts as the additive identity.
• Identity matrix I (or In ): the n × n square matrix with 1s on the leading diagonal and 0s elsewhere.
Acts as the multiplicative identity: AI = IA = A.
1 0 0
1 0
I2 = , I3 = 0 1 0
0 1
0 0 1
• Row matrix: 1 × n. Column matrix: n × 1 (often used for vectors).
• Diagonal matrix: square matrix whose non-diagonal entries are all 0.
1.2 Equality of matrices
Two matrices are equal if and only if they have the same order and every corresponding entry is equal.
Examiner Trap
1 2
A matrix with the same numbers in a different shape is not the same matrix. and
3 4
1 2 3 4 are different matrices: different orders.
2. Matrix Operations
2.1 Addition and subtraction
Two matrices can only be added or subtracted if they have the same order. The operation is performed
element-wise:
(A ± B)ij = aij ± bij
3
Core Pure 1 · Edexcel 9FM0
Matrices
Revision & Exam Pack
Complete specification coverage • Worked examples
Common examiner traps • One-page cheat sheet
Designed for A-Level Further Mathematics students · Edexcel specification
,A-Level Further Maths | Core Pure 1: Matrices Edexcel 9FM0
Specification Map
This pack covers every Matrices learning objective in Edexcel A-Level Further Mathematics
Core Pure 1.
Spec Content Section
6.1 Matrices: addition, subtraction, scalar multiplication, multiplica- §1, §2
tion
6.2 Zero and identity matrices §1
6.3 Linear transformations in 2D §5
6.4 Successive transformations §5
6.5 Invariant points and invariant lines §5
6.6 Determinants (2×2 and 3×3); singular matrices §3
6.7 Inverse matrices (2×2 and 3×3) §4
6.8 Solving three linear simultaneous equations in three variables §6
6.9 Geometrical interpretation of solutions of three planes §6
Quick Tip
Work through this pack actively — pause at each Worked Example and try the problem before
reading the solution. The Examiner Trap boxes pinpoint the mistakes that cost marks in past
papers; learn to recognise them before exam day.
2
, A-Level Further Maths | Core Pure 1: Matrices Edexcel 9FM0
1. Matrix Fundamentals
A matrix is a rectangular array of numbers arranged in rows and columns. We write matrices using
bold capital letters: A, B, M.
The order (or dimensions) of a matrix with m rows and n columns is written m × n. The entry in
the ith row and jth column of A is denoted aij .
Matrix Notation
a11 a12 ··· a1n
a21
a22 ··· a2n
A=
.. .. .. ..
. . . .
am1 am2 · · · amn
A has order m × n. We say “m by n”.
1.1 Special matrices
• Square matrix: same number of rows and columns (n × n).
• Zero matrix 0: every entry is 0. Acts as the additive identity.
• Identity matrix I (or In ): the n × n square matrix with 1s on the leading diagonal and 0s elsewhere.
Acts as the multiplicative identity: AI = IA = A.
1 0 0
1 0
I2 = , I3 = 0 1 0
0 1
0 0 1
• Row matrix: 1 × n. Column matrix: n × 1 (often used for vectors).
• Diagonal matrix: square matrix whose non-diagonal entries are all 0.
1.2 Equality of matrices
Two matrices are equal if and only if they have the same order and every corresponding entry is equal.
Examiner Trap
1 2
A matrix with the same numbers in a different shape is not the same matrix. and
3 4
1 2 3 4 are different matrices: different orders.
2. Matrix Operations
2.1 Addition and subtraction
Two matrices can only be added or subtracted if they have the same order. The operation is performed
element-wise:
(A ± B)ij = aij ± bij
3
