03 మాత్రికలు
వ్యాయామం 04
Question:
Find the inverse of the matrices , if it exists. [1327]
Answer:
Step 1: Calculate the determinant of the matrix.
Determinant of the matrix = 1 × 7 - 3 × 2 = 5
Step 2: Since the determinant is not equal to 0, the inverse of the matrix exists.
Step 3: Calculate the adjoint of the matrix.
Adjoint of the matrix = [7-3-21]
Step 4: Divide the adjoint by the determinant of the matrix.
Inverse of the matrix = [7-3-21]/5
= [7-35-251]
Question:
Find the inverse of the matrices, if it exists. [2513]
Answer:
Step 1: Calculate the determinant of the matrix.
The determinant of the matrix is |2513|=1−15=-13
Step 2: Check if the determinant is not equal to zero.
The determinant is not equal to zero, so the inverse of the matrix exists.
Step 3: Calculate the adjoint of the matrix.
The adjoint of the matrix is [3−5−12]
Step 4: Divide the adjoint matrix by the determinant.
The inverse of the matrix is [3−5−12]/-13]=[3/-13−5/-13−1/-132/-13]=[−3/135/131/13−2/13]
Question:
Find the inverse of the matrices, if it exists. [4534]
Answer:
Step 1: Find the determinant of the matrix.
Determinant of the matrix = 4 × 4 - 5 × 3 = 16 - 15 = 1
Step 2: Since the determinant is not 0, the inverse of the matrix exists.
Step 3: Calculate the adjoint of the matrix.
Adjoint of the matrix = [4-5-34]
Step 4: Divide the adjoint of the matrix by the determinant to get the inverse of the matrix.
Inverse of the matrix = [4-5-34]/1
= [4-5-34]
Question:
Find the inverse of the matrices , if it exists. [2-3-12]
Answer:
Step 1: Calculate the determinant of the matrix.
Determinant = 2 × 2 - (-3) × (-1) = 7
Step 2: Calculate the inverse of the matrix, if it exists.
Inverse = (1/7) × [2-3-12]
= (1/7) × [2-3-12]
= (1/7) × [2-3-12]
= (1/7) × [231-2]
= [2/73/71/7-2/7]
Therefore, the inverse of the given matrix is [2/73/71/7-2/7].
Question:
Find the inverse of the matrices, if it exists. [3-1-42]
Answer:
Step 1: Calculate the determinant of the matrix.
[3-1-42]
Determinant = 32 - (-1)(-4) = 10
Step 2: Check if the determinant is non-zero.
The determinant is 10, which is non-zero. Therefore, the inverse of the matrix exists.
Step 3: Calculate the adjoint of the matrix.
[3-1-42]
Adjoint = [24-13]
Step 4: Divide the adjoint matrix by the determinant.
[24-13]/10
Inverse = [0.20.4-0.10.3]
Question:
Find the inverse of each of the matrices, if it exists. [1-123]
Answer:
Step 1: Calculate the determinant of the matrix.
The determinant of the matrix is 5.
Step 2: Calculate the inverse of the matrix.
The inverse of the matrix is [3-1-21] divided by the determinant (5).
Therefore, the inverse of the matrix is [3-1-21]/5.
Question:
Find the inverse of each of the matrices , if it exists. [2111]
Answer:
Step 1: Calculate the determinant of the matrix.
Determinant = 2 - 1 = 1
Step 2: Since the determinant is not zero, the matrix is invertible.
Step 3: Calculate the adjoint of the matrix.
[1-1-12]
Step 4: Multiply the adjoint by 1/determinant.
[1-1-12] ÷ 1]
Step 5: The inverse of the matrix is
[1-1-12]
Question:
Matrices A and B will be inverse of each other only if A AB=BA B AB=0,BA=I C AB=BA=0 D AB=BA=I
Answer:
A) Matrices A and B will be inverse of each other only if A × B = B × A
B) A × B × A = B × A × B = 0
C) B × A × B = A × B × A = 0
D) A × B × A = B × A × B = I (where I is the identity matrix)
Question:
Find the inverse of the following of the matrices , if it exists. [2-332233-22]
Answer:
Step 1: Calculate the determinant of the matrix.
The determinant of the matrix is 12.
Step 2: If the determinant is not equal to 0, then the inverse of the matrix exists.
Since the determinant is 12, the inverse of the matrix exists.
Step 3: Use the formula to calculate the inverse of the matrix.
The inverse of the matrix is
[11-2-1212-11]
Question:
Find the inverse of the matrices, if it exists. [31027]
Answer:
Step 1: Calculate the determinant of the matrix.
The determinant of the matrix is [31027] = 37 - 102 = 17 - 20 = -3
Step 2: Check if the determinant is non-zero.
Since the determinant is -3, it is non-zero, so the matrix is invertible.
Step 3: Calculate the adjoint of the matrix.
The adjoint of the matrix is [7-10-23]
Step 4: Divide each element of the adjoint matrix by the determinant.
The inverse of the matrix is [7-3-10-3-2-33-3]
Therefore, the inverse of the matrix is [-7102-3].
Question:
Find the inverse of the matrices, if it exists. [2174]
Answer:
Step 1: Calculate the determinant of the matrix:
|2174|=2−7=−5
Step 2: Check to see if the determinant is non-zero. Since the determinant is -5, the inverse of the matrix exists.
Step 3: Find the adjoint of the matrix.
[2174]⊙[4−7−12]=[47−12]
Step 4: Divide each element of the adjoint matrix by the determinant.
[47−12]÷−5
Step 5: The inverse of the matrix is:
Question:
Find the inverse of the matrix, if it exists.
Answer:
Step 1: Calculate the determinant of the matrix.
Determinant = 2 × 1 × 3 + 0 × 0 × (-1) + 5 × (-1) × 0 = 6
Step 2: Check if the determinant is not zero.
Since the determinant is not zero, the matrix is invertible.
Step 3: Find the adjoint of the matrix.
Adjoint =
Step 4: Divide the adjoint by the determinant.
Inverse =
Therefore, the inverse of the given matrix is .
Question:
Find the inverse of the matrices, if it exists.
Answer:
Step 1: Calculate the determinant of the matrix.
Determinant = 2(-2) - (-6) = 4 + 6 = 10
Step 2: Calculate the adjoint of the matrix.
Adjoint =
Step 3: Divide the adjoint by the determinant.
Inverse =
Inverse =
Question:
Find the inverse of the matrices, if it exists.
Answer:
Step 1: Calculate the determinant of the matrix.
Step 2: Check if the determinant is non-zero.
The determinant is not zero, so the matrix is invertible.
Step 3: Calculate the adjugate of the matrix.
Step 4: Divide the adjugate by the determinant.
Step 5: Simplify the fraction.
Step 6: The inverse of the matrix is:
Question:
Find the inverse of the matrices , if it exists.
Answer:
Step 1: Calculate the determinant of the matrix.
Determinant = 27 - 53 = 1
Step 2: Calculate the adjoint of the matrix.
Step 3: Divide each element of the adjoint matrix by the determinant.
/1 =
Step 4: The inverse of the given matrix is:
Question:
Find the inverse of the matrices , if it exists.
Answer:
Step 1: Calculate the determinant of the given matrix.
Determinant = 6 × 1 - (-3) × (-2) = 12 + 6 = 18
Step 2: Calculate the cofactors of each element in the matrix.
Cofactors of 6 = 1 Cofactors of -3 = 2 Cofactors of -2 = -3 Cofactors of 1 = 6
Step 3: Calculate the adjoint of the matrix by taking the transpose of the matrix of cofactors.
Adjoint = [1 2] [-3 6]
Step 4: Divide each element of the adjoint matrix by the determinant (18).
Inverse = [1/18 2/18] [-3/18 6/18]
Question:
Find the inverse of the matrices , if it exists.
Answer:
Step 1: Calculate the determinant of the given matrix.
Determinant = 1*(00) - 3(-52) + (-25*-3) = -25
Step 2: Since the determinant is not 0, the inverse of the matrix exists.
Step 3: Calculate the cofactor matrix.
Cofactor matrix =
Step 4: Calculate the adjoint matrix by taking the transpose of the cofactor matrix.
Adjoint matrix =
Step 5: Divide the adjoint matrix by the determinant of the given matrix.
Inverse matrix =
Question:
Find the inverse of the matrices , if it exists.
Answer:
Step 1: Rewrite the matrix in the form of
Step 2: Calculate the determinant of the matrix.
Step 3: Check if the determinant is non-zero.
The determinant is -6, which is not zero. Therefore, the inverse of the matrix exists.
Step 4: Calculate the inverse of the matrix using the formula
Step 5: Simplify the inverse matrix.