Math 300
Notes for Section 2.2
1. Theorem (Properties of Matrix Addition). If A,B, and Care m๎nmatrices, then:
(a) AC.B CC/D.A CB/ CC.
(b) ACBDBCA.
2. Theorem (Additive Identity and Inverses). Let Omn be the m๎nmatrix whose each entry is 0.
Let ADลaij ๎be any m๎nmatrix and let ๎ADล๎aij ๎. Then:
(a) ACOmn DAand Omn CADA.
(b) AC.๎A/ DOmn and .๎A/ CADOmn.
3. Theorem (Properties of Matrix Multiplication). If A,B, and Care matrices with sizes such
that the given matrix operations are defined, then:
(a) A.BC / D.AB/C .
(b) A.B CC/DAB CAC .
(c) .A CB/C DAC CBC.
4. Theorem (Properties of the Identity Matrix)..IfAis an m๎nmatrix then:
(a) AInDA.
(b) ImADA.
It follows that if Ais a square matrix of order nthen
InADAInDA:
5. Theorem (Properties of Scalar Multiplication). If A,B, and Care matrices with sizes such
that the given matrix operations are defined, and cand dare scalars, then:
(a) .cd /A Dc.dA/.
(b) 1A DA.
(c) c.A CB/ DcA CcB.
(d) .c Cd/A DcA CdA.
(e) c.AB/ D.cA/B DA.cB/.
6. Theorem (Properties of Transposes). If Aand Bare matrices with sizes such that the given
matrix operations are defined, and cis a scalar, then:
(a) .AT/TDA.
(b) .A CB/TDATCBT.
(c) .cA/TDcAT.
(d) .AB/TDBTAT.
