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Questions and Answers
If matrix A is any m x n matrix, which of the following statements is true?
If matrix A is any m x n matrix, which of the following statements is true?
- $AI_m = A$ and $I_n A = A$
- $A + I_m = A$
- $AI_n = A$ and $I_m A = A$ (correct)
- $A + I_n = A$
Which of the following is always true for a square matrix A raised to the power of 0?
Which of the following is always true for a square matrix A raised to the power of 0?
- $A^0 = A$
- $A^0 = 0$
- $A^0$ is undefined
- $A^0 = I_n$ (correct)
Under what condition is the statement $(AB)^p = A^p B^p$ true, where A and B are square matrices and p is a positive integer?
Under what condition is the statement $(AB)^p = A^p B^p$ true, where A and B are square matrices and p is a positive integer?
- Only if AB = BA (correct)
- Always true
- Only if A = B
- Only if A or B is an identity matrix
A matrix A is upper triangular. Which condition must be met for its elements $a_{ij}$?
A matrix A is upper triangular. Which condition must be met for its elements $a_{ij}$?
What condition defines a symmetric matrix A with real entries?
What condition defines a symmetric matrix A with real entries?
What distinguishes a Skew symmetric matrix from other types of matrices?
What distinguishes a Skew symmetric matrix from other types of matrices?
In the context of a correlation matrix, what does the value of an element $\rho(X, Y)$ represent?
In the context of a correlation matrix, what does the value of an element $\rho(X, Y)$ represent?
For any square matrix A, which of the following expressions defines the trace of A, denoted as Tr(A)?
For any square matrix A, which of the following expressions defines the trace of A, denoted as Tr(A)?
If A and B are matrices, and c is a real number, which of the following properties involving the trace function, Tr, is always true?
If A and B are matrices, and c is a real number, which of the following properties involving the trace function, Tr, is always true?
Under what condition is a square matrix A considered invertible (nonsingular)?
Under what condition is a square matrix A considered invertible (nonsingular)?
Given a 2x2 matrix $A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}$, what is the formula to calculate its determinant, det(A)?
Given a 2x2 matrix $A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}$, what is the formula to calculate its determinant, det(A)?
Consider a matrix A. How is the minor $M_{ij}$ defined?
Consider a matrix A. How is the minor $M_{ij}$ defined?
How is the cofactor $A_{ij}$ of an element $a_{ij}$ in a matrix A defined in relation to the minor $M_{ij}$?
How is the cofactor $A_{ij}$ of an element $a_{ij}$ in a matrix A defined in relation to the minor $M_{ij}$?
Which of the following is a valid method for computing the determinant of a matrix A?
Which of the following is a valid method for computing the determinant of a matrix A?
A determinant of a matrix is calculated by expanding along the first row as follows: det(A) = $a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}$. What do the terms $A_{11}$, $A_{12}$, and $A_{13}$ represent?
A determinant of a matrix is calculated by expanding along the first row as follows: det(A) = $a_{11}A_{11} + a_{12}A_{12} + a_{13}A_{13}$. What do the terms $A_{11}$, $A_{12}$, and $A_{13}$ represent?
If two rows of a matrix A are interchanged, resulting in matrix B, how does det(B) relate to det(A)?
If two rows of a matrix A are interchanged, resulting in matrix B, how does det(B) relate to det(A)?
If a row (or column) of a matrix A is multiplied by a scalar k, resulting in a new matrix B, how does det(B) relate to det(A)?
If a row (or column) of a matrix A is multiplied by a scalar k, resulting in a new matrix B, how does det(B) relate to det(A)?
If matrix B is obtained from matrix A by adding a multiple of one row to another row, how does the determinant of B relate to the determinant of A?
If matrix B is obtained from matrix A by adding a multiple of one row to another row, how does the determinant of B relate to the determinant of A?
The matrix A is upper triangular. What can you say about its determinant?
The matrix A is upper triangular. What can you say about its determinant?
If A and B are n × n matrices, how is det(AB) related to det(A) and det(B)?
If A and B are n × n matrices, how is det(AB) related to det(A) and det(B)?
Which of the following is true of a scalar matrix?
Which of the following is true of a scalar matrix?
What is the primary characteristic of an identity matrix?
What is the primary characteristic of an identity matrix?
Which of the following correctly describes a diagonal matrix?
Which of the following correctly describes a diagonal matrix?
What distinguishes a square matrix from a general $m \times n$ matrix?
What distinguishes a square matrix from a general $m \times n$ matrix?
What is the relationship between the determinant of a matrix and its invertibility?
What is the relationship between the determinant of a matrix and its invertibility?
Flashcards
What is a Square Matrix?
What is a Square Matrix?
A matrix where the number of rows equals the number of columns.
What is a Diagonal Matrix?
What is a Diagonal Matrix?
A square matrix where all non-diagonal elements are zero.
What is a Scalar Matrix?
What is a Scalar Matrix?
A diagonal matrix where all diagonal elements are equal.
What is an Identity Matrix?
What is an Identity Matrix?
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What is AP (A to the power of p)?
What is AP (A to the power of p)?
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What is an Upper Triangular Matrix?
What is an Upper Triangular Matrix?
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What is a Lower Triangular Matrix?
What is a Lower Triangular Matrix?
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What is a Symmetric Matrix?
What is a Symmetric Matrix?
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What is a Skew-Symmetric Matrix?
What is a Skew-Symmetric Matrix?
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What is the Trace of a Matrix?
What is the Trace of a Matrix?
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What is an Invertible Matrix?
What is an Invertible Matrix?
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What is the Determinant of a Matrix?
What is the Determinant of a Matrix?
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What is a Minor?
What is a Minor?
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What is a Cofactor?
What is a Cofactor?
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Determinant property 1
Determinant property 1
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Study Notes
- Linear algebra is presented by Mohamed Amir from MSA University.
Square Matrix
- A square matrix has an equal number of rows and columns.
Types of Square Matrices: Diagonal Matrix
- An n x n square matrix A = [aij] is a diagonal matrix if aij = 0 for i ≠ j.
Types of Square Matrices: Scalar Matrix
- A scalar matrix is a diagonal matrix with equal diagonal elements.
Types of Square Matrices: Identity Matrix
- The scalar matrix with all diagonal entries (dii) equal to 1 is the n x n identity matrix In.
- I3 is the 3x3 identity matrix.
Square Matrix Rules
- For any m × n matrix A, AIn = A and ImA = A.
- For a square matrix A, raising it to a positive integer power p is defined as Ap = A * A * ... * A (p factors).
- If A is an n x n matrix, A0 = In.
- AP Aq = AP+q
- (AP)q = APq
- (AB)P = AP BP, but only if AB = BA.
Types of Square Matrices: Lower/Upper Triangular
- An n x n matrix A = aij is upper triangular if aij = 0 for i > j.
- It is lower triangular if aij = 0 for i < j.
- A diagonal matrix is both upper and lower triangular.
Types of Square Matrices: Symmetric Matrices
- A matrix A with real entries is symmetric if AT = A.
- A matrix A with real entries is skew-symmetric if AT = −A.
Symmetric Matrix Application: Correlation Matrix
- ρ (X,Y) = cov (X,Y) / (σX * σY)
- Cov(X,Y) = E[(X-E(X))
- -1 ≤ p (X,Y) ≤ 1
Trace of a Square Matrix
- The trace of an n x n matrix A, denoted Tr(A), is the sum of all elements on the main diagonal.
- Properties of Trace:
- Tr(cA) = c Tr(A), where c is a real number.
- Tr(A + B) = Tr(A) + Tr(B)
- Tr(AB) = Tr(BA)
- Tr(AT) = Tr(A)
- Tr(ATA) ≥ 0
Invertible Matrix
- An n x n matrix A is nonsingular, or invertible, if there exists an n x n matrix B such that AB = BA = In; such a B is the inverse of A.
- Otherwise, A is singular, or noninvertible.
- If B exists such that AB = BA = I2, then B is the inverse of A, or B = A-1.
Determinant of a Matrix
- For a 2x2 matrix A = [[a11, a12], [a21, a22]], the determinant det(A) = a11a22 - a12a21.
- The determinant of matrix A = [[2, -3], [4, 5]] is calculated as (2)(5) - (-3)(4) = 22
Determinant of a Matrix: Minors
- For an n x n matrix A = [aij], Mij is the (n-1) x (n-1) submatrix of A obtained by deleting the i-th row and j-th column of A.
- The determinant det(Mij) is called the minor of aij.
- Ex: Calculating the minor for a specific element in a 3x3 matrix involves finding the determinant of the 2x2 matrix that remains after removing the row and column of that element. The equation is minor(aij) = det(M12) = det([[4,6],[7,2]]) = 8-42 = -34
Determinant of a Matrix: Cofactors
- The cofactor Aij of aij is defined as Aij = (-1)i+j det(Mij).
Determinant of a Matrix Calculation: Expansion
- Expansion along the i-th row: det(A) = ai1Ai1 + ai2Ai2 + ... + ainAin
- Expansion along the j-th column: det(A) = a1jA1j + a2jA2j + ... + anjAnj
Determinant Properties
- det(AT) = det(A)
- If B is generated by interchanging two rows (or columns) in A, i.e., B = Ari ↔ rj, then det(B) = -det(A).
- If B is obtained from A by multiplying a row (or column) of A by a real number k, then det(B) = k det(A).
- If B = bij is obtained from A = aij by adding to each element of the r-th row (or column) of A, k times the corresponding element of the s-th row (or column), where r ≠ s, then det(B) = det(A).
- If a matrix A = [aij] is upper (or lower) triangular, then det(A) = a11 * a22 * ... * ann; that is, the determinant of a triangular matrix is the product of the elements on the main diagonal.
- If A and B are n x n matrices, then det(AB) = det(A) det(B).
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