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IBMathsDPApplications & Interpretation (AI)HLRevision Notes1. Number & AlgebraMatricesOperations with Matrices

Operations with Matrices (DP IB Applications & Interpretation (AI)): Revision Note

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on

Matrix addition & subtraction

How is addition and subtraction performed with matrices?

  • Two matrices of the same order can be added or subtracted

  • Only corresponding elements of the two matrices are added or subtracted

    • bold italic A plus-or-minus bold italic B equals left parenthesis a subscript i j end subscript right parenthesis plus-or-minus left parenthesis b subscript i j end subscript right parenthesis equals left parenthesis a subscript i j end subscript plus-or-minus b subscript i j end subscript right parenthesis

      • e.g. open parentheses table row 1 0 row 2 cell negative 1 end cell end table close parentheses plus open parentheses table row 2 cell negative 1 end cell row 3 1 end table close parentheses equals open parentheses table row 3 cell negative 1 end cell row 5 0 end table close parentheses

  • The resultant matrix is of the same order as the original matrices being added or subtracted

What are the properties of matrix addition?

  • Addition is commutative

    • bold italic A plus bold italic B equals bold italic B plus bold italic A

  • Addition is associative

    • bold italic A plus left parenthesis bold italic B plus bold italic C right parenthesis equals left parenthesis bold italic A plus bold italic B right parenthesis plus bold italic C

  • The zero matrix is the additive identity

    • bold italic A plus bold italic O equals bold italic A for all matrices bold italic A

  • Subtraction is the inverse of addition

    • bold italic O minus bold italic A equals negative bold italic A

    • bold italic A minus bold italic B equals bold italic A plus left parenthesis negative bold italic B right parenthesis

Examiner Tips and Tricks

Make sure that you know how to add and subtract matrices on your GDC for speed or for checking work in an exam!

Worked Example

Consider the matrices bold italic A equals open parentheses table row cell negative 4 end cell 2 row 7 3 row 1 cell negative 5 end cell end table close parentheses, bold italic B equals open parentheses table row 2 6 row 5 cell negative 9 end cell row cell negative 2 end cell cell negative 3 end cell end table close parentheses.

a) Find bold italic A plus bold italic B.

1-7-2-ib-ai-hl-operations-with-matrices-we-1a-solution

b) Find bold italic A minus bold italic B.

1-7-2-ib-ai-hl-operations-with-matrices-we-1b-solution

Matrix multiplication

How do I multiply a matrix by a scalar?

  • Multiply each element in the matrix by the scalar value

    • k bold italic A equals left parenthesis k a subscript i j end subscript right parenthesis

      • e.g. 3 open parentheses table row 2 0 row 1 2 row cell negative 1 end cell 3 end table close parentheses equals open parentheses table row 6 0 row 3 6 row cell negative 3 end cell 9 end table close parentheses

  • The resultant matrix is of the same order as the original matrix

  • Multiplication by a negative scalar changes the sign of each element in the matrix

How do I multiply a matrix by another matrix?

  • You can only multiply two matrices if their orders are compatible

    • bold italic A cross times bold italic B only exists if the number of columns in bold italic A is equal to the number of rows in bold italic B

  • If the order of bold italic A is m cross times n and the order of bold italic B is n cross times p then the order of bold italic A cross times bold italic B is m cross times p

  • To find the element of bold italic A cross times bold italic Bin the row i and the column j

    • multiply each element in the row iof matrix bold italic Awith the corresponding element in the column j of matrix bold italic B

    • find the sum of the products

      • e.g. If bold italic A equals open square brackets table row a b c row d e f end table close square brackets, bold italic B equals open square brackets table row g h row i j row k l end table close square brackets

        • then bold italic A bold italic B equals open square brackets table row cell left parenthesis a g plus b i plus c k right parenthesis end cell cell left parenthesis a h plus b j plus c l right parenthesis end cell row cell left parenthesis d g plus e i plus f k right parenthesis end cell cell left parenthesis d h plus e j plus f l right parenthesis end cell end table close square brackets 

        • then  bold italic B bold italic A equals open square brackets table row cell left parenthesis g a plus h d right parenthesis end cell cell left parenthesis g b plus h e right parenthesis end cell cell left parenthesis g c plus h f right parenthesis end cell row cell left parenthesis i a plus j d right parenthesis end cell cell left parenthesis i b plus j e right parenthesis end cell cell left parenthesis i c plus j f right parenthesis end cell row cell left parenthesis k a plus l d right parenthesis end cell cell left parenthesis k b plus l e right parenthesis end cell cell left parenthesis k c plus l f right parenthesis end cell end table close square brackets

Examiner Tips and Tricks

You might have used a grid to multiply numbers or expand brackets. You can set out matrix multiplication similarly to make it clear which elements need to get multiplied together.

table row cross times cell bottom enclose open parentheses table row a b row c d row e f end table close parentheses end enclose end cell row cell right enclose open parentheses table row 1 2 3 row 4 5 6 end table close parentheses end enclose end cell cell open parentheses table row times times row times times end table close parentheses end cell end table

What are the properties of matrix multiplication?

  • Multiplication is not commutative

    • In general bold italic A bold italic B not equal to bold italic B bold italic A

    • Order matters

  • Multiplication is associative

    • bold italic A left parenthesis bold italic B bold italic C right parenthesis equals left parenthesis bold italic A bold italic B right parenthesis bold italic C

  • Multiplication is distributive with addition

    • bold italic A left parenthesis bold italic B plus bold italic C right parenthesis equals bold italic A bold italic B plus bold italic A bold italic C

    • left parenthesis bold italic A plus bold italic B right parenthesis bold italic C equals bold italic A bold italic C plus bold italic B bold italic C

  • The identity matrix is the multiplicative identity

    • bold italic A bold italic I equals bold italic I bold italic A equals bold italic A

  • Any product involving a zero matrix results in a zero matrix

    • bold italic A bold italic O equals bold italic O bold italic A equals bold italic O

    • The converse is not true

      • i.e. if bold italic A cross times bold italic B equals bold 0 then it is possible that both bold italic A and bold italic B are non-zero

  • Powers of square matrices are repeated multiplication

    • bold italic A squared equals bold italic A bold italic A

    • bold italic A cubed equals bold italic A bold italic A bold italic A

    • etc

Examiner Tips and Tricks

Matrix multiplication works very similarly to multiplication of real numbers. However, be careful with the order. Some results do not work with matrices because of this.

For example, open parentheses bold italic A plus bold italic B close parentheses squared equals bold italic A squared plus bold italic A bold italic B plus bold italic B bold italic A plus bold italic B squared with might be different to bold italic A squared plus 2 bold italic A bold italic B plus bold italic B squared.

Worked Example

Consider the matrices bold italic A equals open square brackets table row 4 2 cell negative 5 end cell row cell negative 3 end cell 8 1 row cell negative 1 end cell cell negative 2 end cell 2 end table close square brackets and bold italic B equals open square brackets table row 5 1 row cell negative 2 end cell 5 row 9 7 end table close square brackets .

a) Find bold italic A bold italic B.

1-7-2-ib-ai-hl-operations-with-matrices-we-2a-solution

b) Explain why you cannot find bold italic B bold italic A.

1-7-2-ib-ai-hl-operations-with-matrices-we-2b-solution

c) Find bold italic A squared.

rn-1-7-matrices

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    Author
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    Naomi C

    Author: Naomi C

    Expertise: Maths Content Creator

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.