Here are the generated flashcards:
### Flashcard 1: Finding Rank of Matrix
**Front:** What is the first step in finding the rank of a matrix?
**Back:** The first step is to convert the given matrix into its echelon form or reduced echelon form using elementary row operations.
### Flashcard 2: Finding Rank of Matrix
**Front:** What is the rank of a matrix?
**Back:** The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
### Flashcard 3: Consistency
**Front:** What is consistency in the context of linear equations?
**Back:** Consistency refers to the property of a system of linear equations having at least one solution.
### Flashcard 4: Consistency
**Front:** How do you check for consistency in a system of linear equations?
**Back:** A system of linear equations is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix.
### Flashcard 5: Cayley Hamilton Theorem
**Front:** What states the Cayley Hamilton Theorem?
**Back:** The Cayley Hamilton Theorem states that every square matrix satisfies its own characteristic equation.
### Flashcard 6: Cayley Hamilton Theorem
**Front:** What is the characteristic equation of a matrix?
**Back:** The characteristic equation of a matrix A is det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix.
### Flashcard 7: Example - Finding Rank of Matrix
**Front:** Find the rank of the matrix
\[
\begin{pmatrix}
2 & 1 & -1 \\
-2 & -1 & 1 \\
2 & 1 & -1 \\
\end{pmatrix}
\]
**Back:** The matrix can be reduced to
\[
\begin{pmatrix}
2 & 1 & -1 \\
0 & 0 & 0 \\
0 & 0 & 0 \\
\end{pmatrix}
\]
which has 1 linearly independent row, so the rank is 1.
### Flashcard 8: Example - Consistency
**Front:** Is the system of equations
2x + y - z = 1,
-x - y/2 + z/2 = -1/2,
-x - y + z = -1
consistent?
**Back:** Convert to augmented matrix and reduce:
\[
\left[
\begin{array}{ccc|c}
2 & 1 & -1 & 1 \\
-1 & -1/2 & 1/2 & -1/2 \\
-1 & -1 & 1 & -1 \\
\end{array}
\right]
\]
After reduction, if the last column does not have a pivot in a row of zeros, then it is consistent. Detailed reduction shows it is consistent.
### Flashcard 9: Example - Cayley Hamilton Theorem
**Front:** Apply the Cayley Hamilton Theorem to a 2x2 matrix
\[
A =
\begin{pmatrix}
1 & 2 \\
3 & 4 \\
\end{pmatrix}
\]
**Back:** The characteristic equation is
det(A - λI) =
\[
\begin{vmatrix}
1-\lambda & 2 \\
3 & 4-\lambda \\
\end{vmatrix}
= (1-\lambda)(4-\lambda) - 6 = 0
\]
which simplifies to λ^2 - 5λ - 2 = 0. By Cayley Hamilton, A^2 - 5A - 2I = 0.
### Flashcard 10: Example - Cayley Hamilton Theorem Application
**Front:** Calculate A^2 for the matrix in Flashcard 9 to verify the Cayley Hamilton Theorem.
**Back:**
\[
A^2 =
\begin{pmatrix}
1 & 2 \\
3 & 4 \\
\end{pmatrix}
\begin{pmatrix}
1 & 2 \\
3 & 4 \\
\end{pmatrix}
=
\begin{pmatrix}
7 & 10 \\
15 & 22 \\
\end{pmatrix}
\]
And
5A + 2I =
\[
5
\begin{pmatrix}
1 & 2 \\
3 & 4 \\
\end{pmatrix}
+
2
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
7 & 10 \\
15 & 22 \\
\end{pmatrix}
\]
Verifying A^2 - 5A - 2I = 0.
These flashcards cover key concepts and examples related to finding the rank of a matrix, consistency of systems of linear equations, and the Cayley Hamilton Theorem. They can be used for studying and reinforcing understanding of these topics in linear algebra.
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Transform any subject into personalized study flashcards instantly.Open472316Released 14d ago100% FreeHere are the generated flashcards: ### Flashcard 1: Finding Rank of Matrix **Front:** What is the first step in finding the rank of a matrix? **Back:** The first step is to convert the given matrix into its echelon form or reduced echelon form using elementary row operations. ### Flashcard 2: Finding Rank of Matrix **Front:** What is the rank of a matrix? **Back:** The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. ### Flashcard 3: Consistency **Front:** What is consistency in the context of linear equations? **Back:** Consistency refers to the property of a system of linear equations having at least one solution. ### Flashcard 4: Consistency **Front:** How do you check for consistency in a system of linear equations? **Back:** A system of linear equations is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix. ### Flashcard 5: Cayley Hamilton Theorem **Front:** What states the Cayley Hamilton Theorem? **Back:** The Cayley Hamilton Theorem states that every square matrix satisfies its own characteristic equation. ### Flashcard 6: Cayley Hamilton Theorem **Front:** What is the characteristic equation of a matrix? **Back:** The characteristic equation of a matrix A is det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix. ### Flashcard 7: Example - Finding Rank of Matrix **Front:** Find the rank of the matrix \[ \begin{pmatrix} 2 & 1 & -1 \\ -2 & -1 & 1 \\ 2 & 1 & -1 \\ \end{pmatrix} \] **Back:** The matrix can be reduced to \[ \begin{pmatrix} 2 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} \] which has 1 linearly independent row, so the rank is 1. ### Flashcard 8: Example - Consistency **Front:** Is the system of equations 2x + y - z = 1, -x - y/2 + z/2 = -1/2, -x - y + z = -1 consistent? **Back:** Convert to augmented matrix and reduce: \[ \left[ \begin{array}{ccc|c} 2 & 1 & -1 & 1 \\ -1 & -1/2 & 1/2 & -1/2 \\ -1 & -1 & 1 & -1 \\ \end{array} \right] \] After reduction, if the last column does not have a pivot in a row of zeros, then it is consistent. Detailed reduction shows it is consistent. ### Flashcard 9: Example - Cayley Hamilton Theorem **Front:** Apply the Cayley Hamilton Theorem to a 2x2 matrix \[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ \end{pmatrix} \] **Back:** The characteristic equation is det(A - λI) = \[ \begin{vmatrix} 1-\lambda & 2 \\ 3 & 4-\lambda \\ \end{vmatrix} = (1-\lambda)(4-\lambda) - 6 = 0 \] which simplifies to λ^2 - 5λ - 2 = 0. By Cayley Hamilton, A^2 - 5A - 2I = 0. ### Flashcard 10: Example - Cayley Hamilton Theorem Application **Front:** Calculate A^2 for the matrix in Flashcard 9 to verify the Cayley Hamilton Theorem. **Back:** \[ A^2 = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ \end{pmatrix} = \begin{pmatrix} 7 & 10 \\ 15 & 22 \\ \end{pmatrix} \] And 5A + 2I = \[ 5 \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ \end{pmatrix} + 2 \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} = \begin{pmatrix} 7 & 10 \\ 15 & 22 \\ \end{pmatrix} \] Verifying A^2 - 5A - 2I = 0. These flashcards cover key concepts and examples related to finding the rank of a matrix, consistency of systems of linear equations, and the Cayley Hamilton Theorem. They can be used for studying and reinforcing understanding of these topics in linear algebra.
