Matrix Operations: Addition, Multiplication, Scalar - Math 300, Sec. 2.2 - Prof. Janusz Ko, Study notes of Linear Algebra

Theorems and properties for matrix addition, multiplication, and scalar multiplication. It covers topics such as associative properties, identity matrices, and distributive properties. These notes are essential for students in a linear algebra or advanced mathematics course.

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Pre 2010

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Math 300
Notes for Section 2.2
1. Theorem (Properties of Matrix Addition). If A,B, and Care m๎˜nmatrices, then:
(a) AC.B CC/D.A CB/ CC.
(b) ACBDBCA.
2. Theorem (Additive Identity and Inverses). Let Omn be the m๎˜nmatrix whose each entry is 0.
Let ADล’aij ๎˜be any m๎˜nmatrix and let ๎˜‚ADล’๎˜‚aij ๎˜. 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 m๎˜nmatrix 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.

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Math 300

Notes for Section 2.

  1. Theorem (Properties of Matrix Addition). If A, B, and C are m  n matrices, then: (a) A C .B C C / D .A C B/ C C. (b) A C B D B C A.
  2. Theorem (Additive Identity and Inverses). Let Omn be the^ mn^ matrix whose each entry is^0. Let A D ล’aij ย be any m  n matrix and let A D ล’aij ย. Then: (a) A C Omn D A and Omn C A D A. (b) A C .A/ D Omn and .A/ C A D Omn.
  3. Theorem (Properties of Matrix Multiplication). If A, B, and C are matrices with sizes such that the given matrix operations are defined, then: (a) A.BC / D .AB/C. (b) A.B C C / D AB C AC. (c) .A C B/C D AC C BC.
  4. Theorem (Properties of the Identity Matrix).. If A is an m  n matrix then: (a) AIn D A. (b) ImA D A. It follows that if A is a square matrix of order n then InA D AIn D A:
  5. Theorem (Properties of Scalar Multiplication). If A, B, and C are matrices with sizes such that the given matrix operations are defined, and c and d are scalars, then: (a) .cd /A D c.dA/. (b) 1A D A. (c) c.A C B/ D cA C cB. (d) .c C d /A D cA C dA. (e) c.AB/ D .cA/B D A.cB/.
  6. Theorem (Properties of Transposes). If A and B are matrices with sizes such that the given matrix operations are defined, and c is a scalar, then: (a) .AT^ /T^ D A. (b) .A C B/T^ D AT^ C BT^. (c) .cA/T^ D cAT^. (d) .AB/T^ D BT^ AT^.