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Unformatted text preview: 3P = 3 −2
1 = 3(−2)
3(1) = −6
3 which corresponds to the point (−6, 3). We can imagine taking (−2, 1) to (−6, 3) in this fashion as
a dilation by a factor of 3 in both the horizontal and vertical directions. Doing this to all points
(x, y ) in the plane, therefore, has the eﬀect of magnifying (scaling) the plane by a factor of 3. 4
5 The Distributive Property, in particular.
See Section 1.8. 8.3 Matrix Arithmetic 479 As did matrix addition, scalar multiplication inherits many properties from real number arithmetic.
Below we summarize these properties.
Theorem 8.4. Properties of Scalar Multiplication
• Associative Property: For every m × n matrix A and scalars k and r, (kr)A = k (rA).
• Identity Property: For all m × n matrices A, 1A = A.
• Additive Inverse Property: For all m × n matrices A, −A = (−1)A.
• Distributive Property of Scalar Multiplication over Scalar Addition: For every
m × n matrix A and scalars k and r,
(k + r)A = kA + rA
• Distributive Property of Scalar Multiplication over Matrix Addition: F...
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