Linear Algebra: Invertible Matrices

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Questions and Answers

Evaluate $1 + 2a + 3b + 2c$ when $a = 3, b = 4, c = 6$

Expand the determinant $\begin{vmatrix} x+y & y+z & z+x \ 1 & 1 & 1 \end{vmatrix}$

In the determinant $\begin{vmatrix} 1 & a & bc \ 1 & b & ca \ 1 & c & ab \end{vmatrix}$, what is the result of subtracting R2 from R1?

$0$

$(b - a)(c - a)$ (correct)

$(a - b)(b - c)(c - a)$

$(a - b)(b - a)$

What is the mathematical term used to refer to the unique number associated with a square matrix?

Determinant Signup and view all the answers

How is the determinant of a square matrix denoted?

| A |, det A, Δ Signup and view all the answers

What is the formula to calculate the determinant of a 2x2 matrix?

a11a22 - a21a12 Signup and view all the answers

How many ways are there to expand a determinant of order 3?

6 Signup and view all the answers

What is the value of the determinant of matrix A = [-1 3 0; 4 1 0]?

-52 Signup and view all the answers

What is the result of applying the operation R1 → R1 + R2 + R3 to the matrix?

abc + bc + ca + ab Signup and view all the answers

What is the result of applying the operation C2 → C2 – C1, C3 → C3 – C1 to the matrix?

1 0 0, b, 1, c Signup and view all the answers

The determinant of a matrix is always a square matrix.

False Signup and view all the answers

What is the correct answer regarding the determinant of a square matrix of order 3 × 3?

k 2 | A | Signup and view all the answers

Define the minor of an element in a determinant.

Minor of an element in a determinant involves deleting the row and column of that element and finding the determinant of the remaining matrix. Signup and view all the answers

What is the minor of the element a11?

3 Signup and view all the answers

What is the cofactor of the element a11?

3 Signup and view all the answers

Find the minor of the element a21.

-4 Signup and view all the answers

Calculate the cofactor of the element a21.

4 Signup and view all the answers

According to the provided content, Delta (Δ) is equal to the sum of the product of elements of any row (or column) with their corresponding ___________.

cofactors Signup and view all the answers

A square matrix A is non-singular if A ≠

0 Signup and view all the answers

Expand the determinant along R1: Δ = 0 - sin α (0 - sin β cos α) - cos α (sin α sin β - 0). Simplify the expression.

sin α sin β cos α - cos α sin α sin β = 0 Signup and view all the answers

Solve the equation 3 - x^2 = 3 - 8 to find the values of x.

x = ±2 Signup and view all the answers

In the matrix A shown, find the determinant |2A| if A = [[1, 2], [4, 1]].

4 Signup and view all the answers

In the matrix A shown, find the determinant |3A| if A = [[1, 0, 1], [0, 1, 2], [0, 0, 4]].

27 Signup and view all the answers

Evaluate the determinant of the matrix: [[3, -1, -2], [0, 0, -1], [3, -4, 5]].

15 Signup and view all the answers

If A = [[1, 2], [2, 1]], what is the value of |A|?

4 Signup and view all the answers

If any two rows (or columns) of a determinant are identical, the value of determinant is nonzero.

False Signup and view all the answers

What happens to the determinant if each element of a row (or column) of a determinant is multiplied by a constant k?

Its value gets multiplied by k. Signup and view all the answers

If AB = BA = I, what is the expression for B in terms of A?

B = (1 / |A|) adj A Signup and view all the answers

For the given matrix A = $\begin{bmatrix} 1 & 3 & 3 \ 4 & 3 & 3 \ 1 & 3 & 4 \end{bmatrix}$, calculate the adjoint of A.

adj A = $\begin{bmatrix} 7 & -3 & -3 \ -1 & 1 & 0 \ -1 & 0 & 1 \end{bmatrix}$ Signup and view all the answers

If A = $\begin{bmatrix} 2 & 3 \ 1 & -4 \end{bmatrix}$ and B = $\begin{bmatrix} -1 & 3 \ 2 & 5 \end{bmatrix}$, what is the result of AB?

$\begin{bmatrix} 5 & -3 \ 5 & 14 \end{bmatrix}$ Signup and view all the answers

For the given system of equations 2x + 5y = 1 and 3x + 2y = 7, what is the solution using matrix method?

x = 3, y = -1 Signup and view all the answers

Solve the following system of equations by matrix method: 3x - 2y + 3z = 8; 2x + y - z = 1; 4x - 3y + 2z = 4

x = 1, y = 2, z = 3 Signup and view all the answers

Represent the system of equations algebraically and find the numbers using matrix method: x + y + z = 6; y + 3z = 11; x - 2y + z = 0

x = 1, y = 2, z = 3 Signup and view all the answers

Examine the consistency of the system of equations: x + 2y = 2; 2x - y = 5; x + 3y = 5; 2x + 3y = 3; x + y = 4; 2x + 6y = 8

Inconsistent systems Signup and view all the answers

Show that the value of the determinant Δ = a b c b c a is negative when a, b, c are positive and unequal.

Negative value Signup and view all the answers

Using properties of determinants, prove: α^2(β + γ) + β^2(γ + α) + γ^2(α + β) = (β - γ)(γ - α)(α - β)(α + β + γ)

Expanding the determinant using properties of determinants yields the desired equality Signup and view all the answers

Using properties of determinants, prove: y^2(1 + py + p^2 z) - z^2(1 + px + p^2 y) = (1 + pxyz)(x - y)(y - z)(z - x)

Expanding the determinant using properties of determinants results in the given equation Signup and view all the answers

Using properties of determinants, prove: -a^2 + b^2 + c^2 - (ab + bc + ac) = 3(a + b + c)(ab + bc + ac)

Expanding the determinant using properties of determinants will lead to the required outcome Signup and view all the answers

Solve the system of equations: 4x - 6y + 5z = 1, 6x - 9y + 20z = 2

(x, y, z) = (2, -1, 4) Signup and view all the answers

If a, b, c are in Arithmetic Progression (A.P.), what is the determinant of the given matrix: [x+2, x+3, x+2a; x+3, x+4, x+2b; x+4, x+5, x+2c]?

0 Signup and view all the answers

For nonzero real numbers x, y, z, what is the inverse of the matrix A = [[x, 0, 0], [0, y, 0], [0, 0, z]]?

[[x^(-1), 0, 0], [0, y^(-1), 0], [0, 0, z^(-1)]] Signup and view all the answers

For the given matrix A = [[1, sinθ, 1], [-sinθ, 1, sinθ], [-1, -sinθ, 1]], what is det(A) when 0 ≤ θ ≤ 2π?

Det(A) = 0 Signup and view all the answers

Prove that the determinant $egin{vmatrix} - ext{sin} heta & x \ ext{cos} heta & 1 \ \ ext{cos} heta & 1 & x \ ext{sin} heta & x \ \ ext{sin} heta & -x & 1 \ ext{cos} heta & -1 & -x \ ext{sin} heta & -1 \ ext{cos} heta & -x \ ext{cos} heta & -x & 1 \ ext{sin} heta & -1 \ ext{cos} heta & x \ ext{sin} heta & x \ ext{cos} heta & 1 \ ext{sin} heta & x \ ext{cos} heta & 1 \ ext{sin} heta & x \ ext{cos} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & -x \ ext{sin} heta & -1 \ ext{cos} heta & -x \ ext{sin} heta & -1 \ ext{cos} heta & -x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{cos} heta & 1 \ ext{sin} heta & x \ ext{cos} heta & 1 \ ext{sin} heta & x \ ext{cos} heta & 1\ ext{sin} heta & x \ ext{cos} heta & 1 \ ext{sin} heta & x \ ext{cos} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & -1 \ ext{cos} heta & -x \ ext{sin} heta & -1 \ ext{cos} heta & -x \ ext{sin} heta & -1 \ ext{cos} heta & -x \ ext{sin} heta & -1 \ ext{cos} heta & -x \ ext{sin} heta & -1 \ ext{cos} heta & -x \ ext{sin} heta & -1 \ ext{cos} heta & -x \ ext{cos} heta & -1 \ ext{sin} heta & -x \ ext{cos} heta & -1 \ ext{sin} heta & -x \ ext{cos} heta & -1 \ ext{sin} heta & -1 \ ext{cos} heta & -x \ ext{sin} heta & -x \ ext{cos} heta & 1 \ ext{sin} heta & x \ ext{cos} heta & 1 \ ext{sin} heta & x \ ext{cos} heta & 1 \ ext{sin} heta & x \ ext{cos} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{cos} heta & -1 \ ext{sin} heta & -x \ ext{cos} heta & -1 \ ext{sin} heta & -x \ ext{cos} heta & -1 \ ext{sin} heta & -x \ ext{cos} heta & -1 \ ext{sin} heta & -x \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x \ ext{sin} heta & 1 \ ext{cos} heta & x

independent of $\theta$ Signup and view all the answers

Prove that $\begin{vmatrix} a & a^2 & bc \ b & b^2 & ca \ c & c^2 & ab \end{vmatrix} = a^2 b^2 c^2$

independent of $\theta$ Signup and view all the answers

Solve the equation $x\frac{{x+a}}{{x}} = 0$, where $a \neq 0$

x = 0, a = -x Signup and view all the answers

Prove that $\begin{vmatrix} a & ab & b^2 \ b & bc & c^2 \ c & ca & a^2 \end{vmatrix} = 4a^2b^2c^2$

4a^2b^2c^2 Signup and view all the answers

Study Notes

Determinants

Determinants are associated with square matrices and can be used to find the solution of a system of linear equations.
A determinant is a number that can be used to determine the uniqueness of a solution of a system of linear equations.

Introduction to Determinants

A determinant is a scalar value that can be used to determine the solvability of a system of linear equations.
Determinants have wide applications in various fields such as Engineering, Science, Economics, and Social Science.

Definition of Determinant

A determinant is a number that can be associated with a square matrix.
The determinant of a square matrix A is denoted by |A| or det A.
The determinant of a 2x2 matrix is defined as:
- |A| = a11a22 - a21a12
The determinant of a 3x3 matrix is defined as:
- |A| = a11(a22a33 - a32a23) - a12(a21a33 - a31a23) + a13(a21a32 - a31a22)

Properties of Determinants

Property 1: The value of the determinant remains unchanged if its rows and columns are interchanged.
Property 2: If any two rows (or columns) of a determinant are interchanged, then the sign of the determinant changes.
These properties are true for determinants of any order.

Evaluating Determinants

There are six ways to expand a 3x3 determinant, corresponding to each of the three rows and three columns.
The expansion of a determinant along a row or column gives the same value.
The value of the determinant can be found by expanding along the row or column that contains the maximum number of zeros.

Examples and Exercises

Evaluate the determinants of given matrices.
Verify the properties of determinants using examples.
Solve exercises to practice evaluating determinants and verifying their properties.### Properties of Determinants
Property 1: If two rows (or columns) of a determinant are identical, then the value of the determinant is zero.
- Example: If R1 and R3 are identical, then Δ = 0.
Property 2: If two rows (or columns) of a determinant are interchanged, then the sign of the determinant is changed.
- Example: If R2 and R3 are interchanged, then Δ = -Δ.
Property 3: If each element of a row (or column) of a determinant is multiplied by a constant k, then the value of the determinant is multiplied by k.
- Example: If each element of the first row is multiplied by k, then Δ1 = kΔ.
Property 4: If some or all elements of a row or column of a determinant are expressed as a sum of two or more terms, then the determinant can be expressed as a sum of two or more determinants.
- Example: a1 + λ1, a2 + λ2, a3 + λ3 = Δ1 + Δ2, where Δ1 and Δ2 are determinants.
Property 5: If to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then the value of the determinant remains the same.
- Example: If Ri → Ri + kRj, then Δ = Δ1.
Property 6: If more than one operation like Ri → Ri + kRj is done in one step, care should be taken to see that a row that is affected in one operation should not be used in another operation.
- Example: Apply R2 → R2 – 2R1 and R3 → R3 – 3R1 to Δ, then Δ = 0.

Examples and Exercises

Example 8: Evaluate Δ = 2a + 3b + 4c. Use Property 5 to show that Δ = 0.
Example 10: Prove that Δ = 4abc. Use Property 5 to show that Δ = 4abc.
Example 12: Show that 1 + xyz = 0. Use Property 5 to show that Δ = 0.
Exercise 4.2: Use properties of determinants to prove the given equations.
Exercise 15: Let A be a square matrix of order 3 × 3, then |kA| is equal to k^3|A|.
Exercise 16: [No question provided]### Determinants
Determinant is a number associated with a square matrix.
The determinant of a matrix can be written in the form of a determinant expression, such as:
- Δ = x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)
The determinant of a matrix is used to find the area of a triangle, where the vertices of the triangle are given by the points (x1, y1), (x2, y2), and (x3, y3).

Area of a Triangle

The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by the expression:
- Δ = x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)
The area of a triangle is always positive, so we take the absolute value of the determinant.
If the area of a triangle is given, we can use both positive and negative values of the determinant for calculation.

Minors and Cofactors

The minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column.
The cofactor of an element aij, denoted by Aij, is defined by:
- Aij = (-1)^(i+j) Mij, where Mij is the minor of aij.
The determinant of a matrix can be expanded along any row or column, and the sum of the products of the elements of the row (or column) with their corresponding cofactors is equal to the determinant.

Adjoint and Inverse of a Matrix

The adjoint of a square matrix A, denoted by adj A, is the transpose of the cofactor matrix.
The adjoint of a matrix A can be used to find the inverse of A, denoted by A^(-1), where A^(-1) = 1/|A| adj A.
A square matrix A is said to be singular if |A| = 0, and nonsingular if |A| ≠ 0.
A nonsingular matrix A is invertible, and its inverse is given by A^(-1) = 1/|A| adj A.
The determinant of the product of matrices is equal to the product of their respective determinants, that is, |AB| = |A| |B|.
If A is a square matrix of order n, then |adj A| = |A|^(n-1).

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This quiz covers the properties of nonsingular matrices, including their invertibility and relation to the identity matrix. It also explores the adjugate matrix and its role in matrix inversion.

Linear Algebra: Invertible Matrices

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Questions and Answers

Evaluate $1 + 2a + 3b + 2c$ when $a = 3, b = 4, c = 6$

Expand the determinant $\begin{vmatrix} x+y & y+z & z+x \ 1 & 1 & 1 \end{vmatrix}$

In the determinant $\begin{vmatrix} 1 & a & bc \ 1 & b & ca \ 1 & c & ab \end{vmatrix}$, what is the result of subtracting R2 from R1?

What is the mathematical term used to refer to the unique number associated with a square matrix?

How is the determinant of a square matrix denoted?

What is the formula to calculate the determinant of a 2x2 matrix?

How many ways are there to expand a determinant of order 3?

What is the value of the determinant of matrix A = [-1 3 0; 4 1 0]?

What is the result of applying the operation R1 → R1 + R2 + R3 to the matrix?

What is the result of applying the operation C2 → C2 – C1, C3 → C3 – C1 to the matrix?

The determinant of a matrix is always a square matrix.

What is the correct answer regarding the determinant of a square matrix of order 3 × 3?

Define the minor of an element in a determinant.

What is the minor of the element a11?

What is the cofactor of the element a11?

Find the minor of the element a21.

Calculate the cofactor of the element a21.

According to the provided content, Delta (Δ) is equal to the sum of the product of elements of any row (or column) with their corresponding ___________.

A square matrix A is non-singular if A ≠

Expand the determinant along R1: Δ = 0 - sin α (0 - sin β cos α) - cos α (sin α sin β - 0). Simplify the expression.

Solve the equation 3 - x^2 = 3 - 8 to find the values of x.

In the matrix A shown, find the determinant |2A| if A = [[1, 2], [4, 1]].

In the matrix A shown, find the determinant |3A| if A = [[1, 0, 1], [0, 1, 2], [0, 0, 4]].

Evaluate the determinant of the matrix: [[3, -1, -2], [0, 0, -1], [3, -4, 5]].

If A = [[1, 2], [2, 1]], what is the value of |A|?

If any two rows (or columns) of a determinant are identical, the value of determinant is nonzero.

What happens to the determinant if each element of a row (or column) of a determinant is multiplied by a constant k?

If AB = BA = I, what is the expression for B in terms of A?

For the given matrix A = $\begin{bmatrix} 1 & 3 & 3 \ 4 & 3 & 3 \ 1 & 3 & 4 \end{bmatrix}$, calculate the adjoint of A.

If A = $\begin{bmatrix} 2 & 3 \ 1 & -4 \end{bmatrix}$ and B = $\begin{bmatrix} -1 & 3 \ 2 & 5 \end{bmatrix}$, what is the result of AB?

For the given system of equations 2x + 5y = 1 and 3x + 2y = 7, what is the solution using matrix method?

Solve the following system of equations by matrix method: 3x - 2y + 3z = 8; 2x + y - z = 1; 4x - 3y + 2z = 4

Represent the system of equations algebraically and find the numbers using matrix method: x + y + z = 6; y + 3z = 11; x - 2y + z = 0

Examine the consistency of the system of equations: x + 2y = 2; 2x - y = 5; x + 3y = 5; 2x + 3y = 3; x + y = 4; 2x + 6y = 8

Show that the value of the determinant Δ = a b c b c a is negative when a, b, c are positive and unequal.

Using properties of determinants, prove: α^2(β + γ) + β^2(γ + α) + γ^2(α + β) = (β - γ)(γ - α)(α - β)(α + β + γ)

Using properties of determinants, prove: y^2(1 + py + p^2 z) - z^2(1 + px + p^2 y) = (1 + pxyz)(x - y)(y - z)(z - x)

Using properties of determinants, prove: -a^2 + b^2 + c^2 - (ab + bc + ac) = 3(a + b + c)(ab + bc + ac)

Solve the system of equations: 4x - 6y + 5z = 1, 6x - 9y + 20z = 2

If a, b, c are in Arithmetic Progression (A.P.), what is the determinant of the given matrix: [x+2, x+3, x+2a; x+3, x+4, x+2b; x+4, x+5, x+2c]?

For nonzero real numbers x, y, z, what is the inverse of the matrix A = [[x, 0, 0], [0, y, 0], [0, 0, z]]?

For the given matrix A = [[1, sinθ, 1], [-sinθ, 1, sinθ], [-1, -sinθ, 1]], what is det(A) when 0 ≤ θ ≤ 2π?

Prove that $\begin{vmatrix} a & a^2 & bc \ b & b^2 & ca \ c & c^2 & ab \end{vmatrix} = a^2 b^2 c^2$

Solve the equation $x\frac{{x+a}}{{x}} = 0$, where $a \neq 0$

Prove that $\begin{vmatrix} a & ab & b^2 \ b & bc & c^2 \ c & ca & a^2 \end{vmatrix} = 4a^2b^2c^2$

Study Notes

Determinants

Introduction to Determinants

Definition of Determinant

Properties of Determinants

Evaluating Determinants

Examples and Exercises

Examples and Exercises

Area of a Triangle

Minors and Cofactors

Adjoint and Inverse of a Matrix

Studying That Suits You

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