CS223 · Annotated SlidesChapter 2 — The Inverse of a Matrix
Chapter 2 · Annotated slides

The Inverse of a Matrix

The 2×2 inverse formula, solving Ax = b with A⁻¹, properties of inverses, elementary matrices, the algorithm [A | I] ~ [I | A⁻¹], the Invertible Matrix Theorem and invertible linear transformations.

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Slide 1: The inverse of a Matrix
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    Slide 3: The inverse of a Matrix
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      Slide 4: The inverse of a Matrix
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      1. (A⁻¹)⁻¹ = A, (AB)⁻¹ = B⁻¹A⁻¹, (Aᵀ)⁻¹ = (A⁻¹)ᵀ.
      Exam trap(AB)⁻¹ = B⁻¹A⁻¹: the order reverses.
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        R3 = R3 + 4R1

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        1. Algorithm: row reduce [A | I]. If A reduces to I, the right block is A⁻¹; otherwise A is not invertible.
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          Slide 8: Characterization of Invertible Matrices
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          1. Invertible Matrix Theorem (n×n A): A invertible ⇔ n pivots ⇔ A ~ I ⇔ Ax = 0 has only the trivial solution ⇔ columns linearly independent ⇔ columns span ℝⁿ ⇔ Ax = b has a unique solution for every b ⇔ det A ≠ 0 ⇔ 0 is not an eigenvalue ⇔ rank A = n.
          TipInvertible Matrix Theorem (n×n A): A invertible ⇔ n pivots ⇔ A ~ I ⇔ Ax = 0 has only the trivial solution ⇔ columns linearly independent ⇔ columns span ℝⁿ ⇔ Ax = b has a unique solution for every b ⇔ det A ≠ 0 ⇔ 0 is not an eigenvalue ⇔ rank A = n.
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          True / False

          If A is n×n and Ax = 0 has a nontrivial solution, then A is invertible.

          Slide 9: Slide 9
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            Slide 10: Invertible Linear Transformations
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