Matrices · 2.2.4
Identity matrix and its properties
Students describe the identity matrix, a square matrix with 1s along the main diagonal and 0s everywhere else, and explain how it behaves under multiplication: multiplying any matrix A by the identity matrix I of matching order leaves A unchanged, so AI = IA = A, similar to multiplying a number by 1.
The official learning standard (2.2.4)
“Explain the characteristics of identity matrix.”
What it means
Students describe the identity matrix, a square matrix with 1s along the main diagonal and 0s everywhere else, and explain how it behaves under multiplication: multiplying any matrix A by the identity matrix I of matching order leaves A unchanged, so AI = IA = A, similar to multiplying a number by 1.
How it is examined
This is usually tested in Paper 1 with a short question asking students to write down the 2×2 identity matrix or verify that AI equals A, and in Paper 2 as a supporting step within larger matrix questions, such as confirming AA⁻¹ = I after finding an inverse matrix.
Worked example
Show that AI = A for A = [[3, 2], [1, 4]] and the 2×2 identity matrix I.
- Write I = [[1, 0], [0, 1]].
- Compute row 1: (3×1 + 2×0), (3×0 + 2×1) = 3, 2
- Compute row 2: (1×1 + 4×0), (1×0 + 4×1) = 1, 4
- So AI = [[3, 2], [1, 4]], which is exactly A.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Is the identity matrix always 2×2 in SPM?
In SPM Form 5, the identity matrix you work with is almost always 2×2, since the chapter focuses on 2×2 matrices. In general, an identity matrix can be any size, but it must always be a square matrix with 1s on the diagonal.
What is the identity matrix used for?
It behaves like the number 1 in matrix multiplication, leaving any matrix unchanged when multiplied by it. It is essential for defining the inverse matrix, since A and its inverse A⁻¹ must multiply together to give the identity matrix I.
Does it matter if I multiply AI or IA?
No, for the identity matrix the order does not matter: AI = IA = A, as long as the orders of A and I are compatible for multiplication. This is a special property that most matrix pairs do not share.