Wize University Linear Algebra Textbook > Matrices

Invertible Matrix Theorem

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Invertible Matrix Theorem
If An×nA_{n \times n}An×n​ is a square matrix, then the following statements are equivalent:
AAA is invertible (non-singular)
ATA^TAT is invertible
rank(A)=n{\rm rank}(A) = nrank(A)=n (AAA has full rank)
The RREF of An×nA_{n \times n}An×n​ is InI_nIn​
AAA is a product of elementary matrices.
The linear system Ax⃗=b⃗A\vec{x}=\vec{b}Ax=b has a unique solution
The homogenous system Ax⃗=0⃗A\vec{x}=\vec{0}Ax=0 has only the trivial solution
det(A)≠0{\rm det}(A)\ne0det(A)=0 (don't worry if you haven't seen this yet!)

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Example: Invertible Matrix Theorem
Consider the linear system 𝐴𝑥⃗=𝑏⃗𝐴𝑥⃗=\vec{𝑏}Ax⃗=b where A5×5A_{5 \times 5}A5×5​ is an invertible matrix. What is rank([𝐴 ∣ b⃗]){\rm rank}([𝐴\ |\ \vec{b}])rank([A ∣ b])?

Since AAA is invertible, AAA has full rank: rank(A)=5{\rm rank}(A)=5rank(A)=5.
Since AAA is full rank, the RREF of AAA is I5I_5I5​ and there is no room left for any more leading 1s.
Therefore, the augmented matrix has the same rank as AAA:  rank(A)=rank([A ∣ b⃗])=5{\rm rank}(A)={\rm rank}([A \ |\ \vec b])=\boxed{5}rank(A)=rank([A ∣ b])=5​.

Suppose that M3×3M_{3 \times 3}M3×3​ is a matrix that can be row reduced into I3I_3I3​.
Which of the following statements are always true? [Select all that apply]

A) MMM is invertible

B) The rank of MMM is less than 3

C) The homogeneous system with coefficient matrix MMM has at least one non-trivial solution

D) The system of linear equations Mx⃗=b⃗M\vec x=\vec bMx=b has infinitely many solutions

I don't know

Extra Practice

Invertible Matrix Theorem

Suppose that M3×3M_{3 \times 3}M3×3​ is a matrix that can be row reduced into I3I_3I3​.
Which of the following statements are always true? [Select all that apply]

Invertible Matrix Theorem

Suppose that M3×3M_{3 \times 3}M3×3​ is a matrix that can be row reduced into I3I_3I3​.
Which of the following statements are always true? [Select all that apply]

Invertible Matrix Theorem

Suppose that M3×3M_{3 \times 3}M3×3​ is a matrix that can be row reduced into I3I_3I3​.
Which of the following statements are always true? [Select all that apply]

Invertible Matrix Theorem

Suppose that M3×3M_{3 \times 3}M3×3​ is a matrix that can be row reduced into I3I_3I3​.
Which of the following statements are always true? [Select all that apply]

Invertible Matrix Theorem

Practice: Invertible Matrix Theorem
Suppose that M3×3M_{3 \times 3}M3×3​ is a matrix that can be row reduced into I3I_3I3​.
Which of the following statements are always true? [Select all that apply]

Let Ax=0A\bm{x}=\bm{0}Ax=0 be a homogeneous system of 5 linear equations in 12 unknowns. If AAA has rank 4. which of the following statements must be false?

Invertible matrix theorem

Suppose that AAA is a 20×2020\times2020×20 coefficient matrix for the homogeneous system of linear equations Ax⃗=0⃗A\vec{x}=\vec{0}Ax=0, which only has the trivial solution. Which of the following statements must be true?
(Select all that are correct)

Invertible Matrix Theorem

Which of the following statements is/are true?
(Select all that are true)

Wize University Linear Algebra Textbook > Matrices

Invertible Matrix Theorem

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Invertible Matrix Theorem

Example: Invertible Matrix Theorem

Extra Practice

Invertible Matrix Theorem

Invertible Matrix Theorem

Invertible Matrix Theorem

Invertible Matrix Theorem

Invertible Matrix Theorem

Invertible matrix theorem

Invertible Matrix Theorem