Determinants - What Is Inverse Of A Matrix

Determinants - What is Inverse of a Matrix

A matrix is invertible or non-singular if and only if its determinant is non-zero.
The inverse of a matrix A is denoted as A^-1. It is the reciprocal of the determinant of A times the adjoint of A.
The inverse of a matrix A can be found using the formula: A^-1 = (1/det(A)) * adj(A)
The adjoint of a matrix A is obtained by taking the transpose of the cofactor matrix of A.
If A is an invertible matrix, then the equation AX = B has a unique solution X, which is given by X = A^-1B.

Finding Inverse of a 2x2 Matrix

Let’s consider a 2x2 matrix A = [[a, b], [c, d]]
The determinant of A is given by det(A) = ad - bc
If the determinant of A is non-zero, then A has an inverse.
The inverse of A is given by A^-1 = [[d, -b], [-c, a]] / (ad - bc)
Example: For matrix A = [[2, 3], [4, 5]], the determinant is 25 - 34 = 10 - 12 = -2. Since the determinant is non-zero, A has an inverse.
Therefore, A^-1 = [[5, -3], [-4, 2]] / -2 = [[-5/2, 3/2], [2, -1]]

Properties of Inverse

If A is an invertible matrix, then (A^-1)^-1 = A
If A and B are invertible matrices, then AB is also invertible and (AB)^-1 = B^-1A^-1
If A is invertible, then the transpose of A, denoted as A^T, is also invertible and (A^T)^-1 = (A^-1)^T
If A is invertible and k is a non-zero constant, then kA is invertible and (kA)^-1 = (1/k)A^-1
Example: Let A = [[2, 1], [-1, 3]] and B = [[4, -2], [3, 5]] be invertible matrices. The inverse of AB is given by (AB)^-1 = B^-1A^-1
Therefore, (AB)^-1 = [[5, -1], [7, -8]]*[[2, 1], [-1, 3]] = [[19, -11], [-7, 8]]

Solving Equations using Matrix Inverse

Inverse matrices can be used to solve systems of linear equations.
Consider the system of equations AX = B, where A is an invertible matrix, X is a vector of variables, and B is a vector of constants.
To find the solution for X, we can multiply both sides of the equation by A^-1: A^-1AX = A^-1B
Simplifying the equation, we get X = A^-1B
Example: Solve the system of equations:
- 2x + 3y = 8
- 4x + 5y = 13
Writing the system in matrix form: AX = B, where A = [[2, 3], [4, 5]], X = [[x], [y]], and B = [[8], [13]]
The inverse of matrix A is A^-1 = [[5, -3], [-4, 2]]
Multiplying both sides of the equation by A^-1, we get X = A^-1B = [[5, -3], [-4, 2]] * [[8], [13]] = [[-19], [12]]
Therefore, the solution to the system of equations is x = -19, y = 12.

Properties of Determinants

The determinant of a matrix is a scalar value which has several important properties.
If A is a square matrix, then the determinant of A is denoted as |A| or det(A).
If A and B are square matrices of the same order, then |AB| = |A| * |B|
If A is a square matrix and k is a scalar, then |kA| = k^n * |A|, where n is the order of matrix A.
If A is invertible, then |A^-1| = 1/|A|
If A is a square matrix and A^T is its transpose, then |A^T| = |A|

Properties of Determinants (Continued)

If A is a square matrix, then |A| = 0 if and only if A is non-invertible or singular.
If A is a triangular matrix, then |A| is equal to the product of its diagonal entries.
If A is a square matrix, then |A| = |A^T|
If A is a square matrix and B is obtained by performing an elementary row operation on A, then |B| = |A|
If A is a square matrix and B is obtained by interchanging two rows (or columns) of A, then |B| = -|A|
Example: Let A = [[2, 3], [4, 6]] be a square matrix. The determinant of A is 26 - 34 = 12 - 12 = 0. Therefore, A is a singular matrix.

Determinants - What is Inverse of a Matrix

A matrix is invertible or non-singular if and only if its determinant is non-zero.
The inverse of a matrix A is denoted as A^-1. It is the reciprocal of the determinant of A times the adjoint of A.
The inverse of a matrix A can be found using the formula: A^-1 = (1/det(A)) * adj(A)
The adjoint of a matrix A is obtained by taking the transpose of the cofactor matrix of A.
If A is an invertible matrix, then the equation AX = B has a unique solution X, which is given by X = A^-1B.

Finding Inverse of a 2x2 Matrix

Let’s consider a 2x2 matrix A = [[a, b], [c, d]]
The determinant of A is given by det(A) = ad - bc
If the determinant of A is non-zero, then A has an inverse.
The inverse of A is given by A^-1 = [[d, -b], [-c, a]] / (ad - bc)
Example: For matrix A = [[2, 3], [4, 5]], the determinant is 25 - 34 = 10 - 12 = -2. Since the determinant is non-zero, A has an inverse.
Therefore, A^-1 = [[5, -3], [-4, 2]] / -2 = [[-5/2, 3/2], [2, -1]]

Properties of Inverse

If A is an invertible matrix, then (A^-1)^-1 = A
If A and B are invertible matrices, then AB is also invertible and (AB)^-1 = B^-1A^-1
If A is invertible, then the transpose of A, denoted as A^T, is also invertible and (A^T)^-1 = (A^-1)^T
If A is invertible and k is a non-zero constant, then kA is invertible and (kA)^-1 = (1/k)A^-1
Example: Let A = [[2, 1], [-1, 3]] and B = [[4, -2], [3, 5]] be invertible matrices. The inverse of AB is given by (AB)^-1 = B^-1A^-1
Therefore, (AB)^-1 = [[5, -1], [7, -8]]*[[2, 1], [-1, 3]] = [[19, -11], [-7, 8]]

Solving Equations using Matrix Inverse

Inverse matrices can be used to solve systems of linear equations.
Consider the system of equations AX = B, where A is an invertible matrix, X is a vector of variables, and B is a vector of constants.
To find the solution for X, we can multiply both sides of the equation by A^-1: A^-1AX = A^-1B
Simplifying the equation, we get X = A^-1B
Example: Solve the system of equations:
- 2x + 3y = 8
- 4x + 5y = 13
Writing the system in matrix form: AX = B, where A = [[2, 3], [4, 5]], X = [[x], [y]], and B = [[8], [13]]
The inverse of matrix A is A^-1 = [[5, -3], [-4, 2]]
Multiplying both sides of the equation by A^-1, we get X = A^-1B = [[5, -3], [-4, 2]] * [[8], [13]] = [[-19], [12]]
Therefore, the solution to the system of equations is x = -19, y = 12.

Determinants - What is Inverse of a Matrix

Properties of Determinants

If A is a square matrix, then |A| = 0 if and only if A is non-invertible or singular.
If A is a triangular matrix, then |A| is equal to the product of its diagonal entries.
If A is a square matrix, then |A| = |A^T|
If A is a square matrix and B is obtained by performing an elementary row operation on A, then |B| = |A|
If A is a square matrix and B is obtained by interchanging two rows (or columns) of A, then |B| = -|A|
Example: Let A = [[2, 3], [4, 6]] be a square matrix. The determinant of A is 26 - 34 = 12 - 12 = 0. Therefore, A is a singular matrix.

Finding the Inverse of a 3x3 Matrix

Let’s consider a 3x3 matrix A = [[a, b, c], [d, e, f], [g, h, i]]
The determinant of A is given by det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
If the determinant of A is non-zero, then A has an inverse.
The inverse of A is given by A^-1 = (1/det(A)) * adj(A), where adj(A) is the adjoint of A.
The adjoint of A is obtained by taking the cofactor matrix of A and then taking its transpose.
Example: For matrix A = [[2, 3, 4], [1, -1, 2], [3, 2, -1]], the determinant is 2(2*(-1) - 22) - 3(1(-1) - 23) + 4(12 - (-1)*3) = 2(-4 - 4) - 3(-1 - 6) + 4(2 + 3) = -16 + 21 + 20 = 25. Since the determinant is non-zero, A has an inverse.
Therefore, A^-1 = (1/25) * adj(A)

Finding the Inverse of a 3x3 Matrix (Continued)

To find the adjoint of A, we need to find the cofactor matrix of A and then take its transpose.
The cofactor matrix of A is obtained by taking the determinant of each minor of A and attaching the sign depending on the position of the element in A.
Let’s find the cofactor matrix of A:
- C = [[-6, 17, -9], [-2, -10, 3], [5, 14, -11]]
The adjoint of A is obtained by taking the transpose of C:
- adj(A) = [[-6, -2, 5], [17, -10, 14], [-9, 3, -11]]
Therefore, A^-1 = (1/25) * [[-6, -2, 5], [17, -10, 14], [-9, 3, -11]]

Finding the Inverse of a 3x3 Matrix (Example)

Example: Let A = [[2, 3, 4], [1, -1, 2], [3, 2, -1]] be a 3x3 matrix.
The determinant of A is det(A) = 25
The adjoint of A is adj(A) = [[-6, -2, 5], [17, -10, 14], [-9, 3, -11]]
Therefore, A^-1 = (1/25) * [[-6, -2, 5], [17, -10, 14], [-9, 3, -11]]
Example: Solve the system of equations using matrix inverse:
- 2x + 3y + 4z = 10
- x - y + 2z = 5
- 3x + 2y - z = 3
Writing the system in matrix form: AX = B, where A = [[2, 3, 4], [1, -1, 2], [3, 2, -1]], X = [[x], [y], [z]], and B = [[10], [5], [3]]
The inverse of matrix A is A^-1 = [[-6/25, -2/25, 5/25], [17/25, -10/25, 14/25], [-9/25, 3/25, -11/25]]
Multiplying both sides of the equation by A^-1, we get X = A^-1B = [[-6/25, -2/25, 5/25], [17/25, -10/25, 14/25], [-9/25, 3/25, -11/25]] * [[10], [5], [3]] = [[2/5], [1/5], [-1/5]]
Therefore, the solution to the system of equations is x = 2/5, y = 1/5, z = -1/5.

Properties of Inverse (Continued)

If A is an invertible matrix and B is a non-zero matrix, then AB = 0 has only the trivial solution.
If A is invertible and AB = AC, where B and C are non-zero matrices, then B = C.
If A and B are invertible matrices, then (AB)^-1 = B^-1A^-1.
If A is an invertible matrix, then (A^T)^-1 = (A^-1)^T.
If A is invertible and symmetric, then A^-1 is also symmetric.

Determinants and Cramer’s Rule

Cramer’s Rule is a method to solve systems of linear equations using determinants.
Consider a system of n linear equations in n variables: AX = B, where A is a square matrix, X is a vector of variables, and B is a vector of constants.
If the determinant of A is non-zero, then the system has a unique solution, given by X = (1/|A|) * C, where C is a vector of determinants.
The i-th determinant in vector C is obtained by replacing the i-th column of A with the vector B.
Example: Solve the system of equations using Cramer’s Rule:
- 2x + y = 5
- x - y = 1
Writing the system in matrix form: AX = B, where A = [[2, 1], [1, -1]], X = [[x], [y]], and B = [[5], [1]]
The determinant of A is |A| = (2*(-1) - 1*1) = (-2 - 1) = -3
The determinant of X with the first column replaced by B is |X1| = [[5, 1], [1, -1]] = (5*(-1) - 1*1) = (-5 - 1) = -6
The determinant of X with the second column replaced by B is |X2| = [[2, 5], [1, 1]] = (21 - 51) = (2 - 5) = -3
Applying Cramer’s Rule, we get x = |X1| / |A| = -6 / -3 = 2 and y = |X2| / |A| = -3 / -3 = 1
Therefore, the solution to the system of equations is x = 2, y = 1.