Properties of Matrix Operations
The main properties of matrix operations such as addition, multiplication, transpose and inverse are presented.
In what follows, \( A, B \) and \( C \) are matrices whose sizes are such that the operations are well defined and \( k \) and \( m \) are scalars.
\( I_n \) is the identity matrix of size \( n \times n \) whose diagonal entries are all equal to 1 and all non diagonal entries equal to zero.
\( 0 \) is the zero matrix whose entries are all zeros.
Properties of Matrix Addition
- \( A + 0 = 0 + A = A\) , where \( 0 \) is the zero matrix.
- \( A + B = B + A \)
- \( (A + B) + C = A + (B + C) \)
Properties of Matrix Multiplication
- \( (A B) C = B(A C) \)
- \( A I_n = A \) , where \( I_n \) is the identity matrix.
- \( I_n A = A \)
- \( 0 A = 0 \) , where \( 0 \) is the zero matrix.
- In general \( AB \ne BA \)
Distributive Properties of Matrices
- \( A(B \pm C) = AB \pm AC \)
- \( (A \pm B)C = AC \pm BC \)
Properties of Matrix Multiplication by Scalars
- \( k(A \pm B) = k A \pm k B \)
- \( (k \pm m)A = kA \pm mA \)
- \( k(m A) = (k m)A \)
- \( k(AB) = (kA)B = A(kB) \)
Properties of Matrix Transpose
- \( (A^T)^T = A \)
- \( (A \pm B)^T = A^T \pm B^T \)
- \( ( k A)^T = k A^T \)
- \( (A B)^T = B^T A^T \)
- \( (I_n)^T = I_n \)
Properties of Matrix Inverse
- \( A A^{-1} = A A^{-1} = I_n \)
- \( (A^{-1})^{-1} = A \)
- \( (k A)^{-1} = k^{-1} A^{-1} = \dfrac{1}{k} A^{-1} \) , for \( k \ne 0 \)
- \( (A B)^{-1} = B^{-1} A^{-1} \)
- \( I_n^{-1} = I_n \)
- \( (A^T)^{-1} = (A^{-1})^T \)
More References and links
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Linear Algebra - Questions with Solutions
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Matrices with Examples and Questions with Solutions.
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Determinant of a Square Matrix.
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Inverse Matrix Questions with Solutions.
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Transpose of a Matrix.
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orthogonal matrix
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orthonormal
- Elementary Linear Algebra - 7 th Edition - Howard Anton and Chris Rorres
- Introduction to Linear Algebra - Fifth Edition (2016) - Gilbert Strang
- Linear Algebra Done Right - third edition, 2015 - Sheldon Axler
- Linear Algebra with Applications - 2012 - Gareth Williams