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Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M–1 = adj (adj M), then which of the following statement is/are ALWAYS TRUE ?
- a)M = I
- b)det M = 1
- c)M2 = I
- d)(adj M)2 = I
Correct answer is option 'B,C,D'. Can you explain this answer?
Most Upvoted Answer
Let M be a 3 × 3 invertible matrix with real entries and let I d...
det (M) ≠ 0
M–1 = adj(adj M)
M–1 = det(M).M
M–1M = det(M).M2
I = det(M).M2 …. (i)
det(I) = (det(M))5
1 = det(M) …. (ii)
From (i) I = M2
(adj M)2 = adj (M2) = adj I = I
M–1 = adj(adj M)
M–1 = det(M).M
M–1M = det(M).M2
I = det(M).M2 …. (i)
det(I) = (det(M))5
1 = det(M) …. (ii)
From (i) I = M2
(adj M)2 = adj (M2) = adj I = I
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Let M be a 3 × 3 invertible matrix with real entries and let I d...
X 2 matrix and N be a 2 x 4 matrix. In order to multiply these matrices, the number of columns in M must be equal to the number of rows in N. Since M has 2 columns and N has 2 rows, we can multiply these matrices.
To find the resulting matrix, we will use the formula for matrix multiplication. Let R be the resulting matrix, then:
R(i,j) = sum(M(i,k) * N(k,j)), where k goes from 1 to 2.
Therefore, R will be a 3 x 4 matrix. Each element in R is found by multiplying the corresponding row in M by the corresponding column in N, and taking the sum of the products. Here is the resulting matrix:
R = [ M(1,1) * N(1,1) + M(1,2) * N(2,1) , M(1,1) * N(1,2) + M(1,2) * N(2,2) , M(1,1) * N(1,3) + M(1,2) * N(2,3) , M(1,1) * N(1,4) + M(1,2) * N(2,4)
M(2,1) * N(1,1) + M(2,2) * N(2,1) , M(2,1) * N(1,2) + M(2,2) * N(2,2) , M(2,1) * N(1,3) + M(2,2) * N(2,3) , M(2,1) * N(1,4) + M(2,2) * N(2,4)
M(3,1) * N(1,1) + M(3,2) * N(2,1) , M(3,1) * N(1,2) + M(3,2) * N(2,2) , M(3,1) * N(1,3) + M(3,2) * N(2,3) , M(3,1) * N(1,4) + M(3,2) * N(2,4) ]
Note that each element in the resulting matrix is a sum of two products, where the first product comes from the corresponding row in M and the second product comes from the corresponding column in N.
To find the resulting matrix, we will use the formula for matrix multiplication. Let R be the resulting matrix, then:
R(i,j) = sum(M(i,k) * N(k,j)), where k goes from 1 to 2.
Therefore, R will be a 3 x 4 matrix. Each element in R is found by multiplying the corresponding row in M by the corresponding column in N, and taking the sum of the products. Here is the resulting matrix:
R = [ M(1,1) * N(1,1) + M(1,2) * N(2,1) , M(1,1) * N(1,2) + M(1,2) * N(2,2) , M(1,1) * N(1,3) + M(1,2) * N(2,3) , M(1,1) * N(1,4) + M(1,2) * N(2,4)
M(2,1) * N(1,1) + M(2,2) * N(2,1) , M(2,1) * N(1,2) + M(2,2) * N(2,2) , M(2,1) * N(1,3) + M(2,2) * N(2,3) , M(2,1) * N(1,4) + M(2,2) * N(2,4)
M(3,1) * N(1,1) + M(3,2) * N(2,1) , M(3,1) * N(1,2) + M(3,2) * N(2,2) , M(3,1) * N(1,3) + M(3,2) * N(2,3) , M(3,1) * N(1,4) + M(3,2) * N(2,4) ]
Note that each element in the resulting matrix is a sum of two products, where the first product comes from the corresponding row in M and the second product comes from the corresponding column in N.
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Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M–1 = adj (adj M), then which of the following statement is/are ALWAYS TRUE ? a)M = Ib)det M = 1c)M2 = Id)(adj M)2 = ICorrect answer is option 'B,C,D'. Can you explain this answer?
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Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M–1 = adj (adj M), then which of the following statement is/are ALWAYS TRUE ? a)M = Ib)det M = 1c)M2 = Id)(adj M)2 = ICorrect answer is option 'B,C,D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M–1 = adj (adj M), then which of the following statement is/are ALWAYS TRUE ? a)M = Ib)det M = 1c)M2 = Id)(adj M)2 = ICorrect answer is option 'B,C,D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M–1 = adj (adj M), then which of the following statement is/are ALWAYS TRUE ? a)M = Ib)det M = 1c)M2 = Id)(adj M)2 = ICorrect answer is option 'B,C,D'. Can you explain this answer?.
Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M–1 = adj (adj M), then which of the following statement is/are ALWAYS TRUE ? a)M = Ib)det M = 1c)M2 = Id)(adj M)2 = ICorrect answer is option 'B,C,D'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M–1 = adj (adj M), then which of the following statement is/are ALWAYS TRUE ? a)M = Ib)det M = 1c)M2 = Id)(adj M)2 = ICorrect answer is option 'B,C,D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let M be a 3 × 3 invertible matrix with real entries and let I denote the 3 × 3 identity matrix. If M–1 = adj (adj M), then which of the following statement is/are ALWAYS TRUE ? a)M = Ib)det M = 1c)M2 = Id)(adj M)2 = ICorrect answer is option 'B,C,D'. Can you explain this answer?.
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