System of Equations With Matrices: Background You’ll Need 2

  • Use the identity and inverse properties of numbers to solve math problems.

Identity Properties

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number.

 [latex]a+0=a[/latex]

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

[latex]a\cdot 1=a[/latex]
  • [latex]\left(-6\right)+0=-6[/latex]
  • [latex]23\cdot 1=23[/latex]

Note: There are no exceptions for these properties; they work for every real number, including [latex]0[/latex] and [latex]1[/latex].

Inverse Properties

The inverse property of addition states that, for every real number [latex]a[/latex], there is a unique number, called the additive inverse (or opposite), denoted [latex]−a[/latex], that, when added to the original number, results in the additive identity, [latex]0[/latex].

[latex]a+\left(-a\right)=0[/latex]

The inverse property of multiplication holds for all real numbers except [latex]0[/latex] because the reciprocal of [latex]0[/latex] is not defined. The property states that, for every real number [latex]a[/latex], there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\frac{1}{a}[/latex], that, when multiplied by the original number, results in the multiplicative identity, [latex]1[/latex].

[latex]a\cdot \dfrac{1}{a}=1[/latex]
  • If [latex]a=-8[/latex], the additive inverse is [latex]8[/latex], since [latex]\left(-8\right)+8=0[/latex].
  • If [latex]a=-\frac{2}{3}[/latex], the reciprocal, denoted [latex]\frac{1}{a}[/latex], is [latex]-\frac{3}{2}[/latex] because
    [latex]a\cdot \dfrac{1}{a}=\left(-\dfrac{2}{3}\right)\cdot \left(-\dfrac{3}{2}\right)=1[/latex]