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The product of two orthogonal matrices is a ... matrix
- a)Unitary
- b)Unit
- c)Orthogonal
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
The product of two orthogonal matrices is a ... matrixa)Unitaryb)Unitc...
Explanation:
Two matrices are orthogonal if their product is the identity matrix.
Let's assume that we have two orthogonal matrices, A and B, and we want to find the product of these matrices.
Product of two matrices:
The product of two matrices is obtained by multiplying the corresponding entries of the matrices and adding them up.
If A is an m×n matrix and B is an n×p matrix, then the product of A and B is an m×p matrix.
Orthogonal matrices:
A matrix A is orthogonal if its transpose is equal to its inverse.
A matrix A is orthogonal if A^T * A = I, where I is the identity matrix.
If A and B are orthogonal matrices, then (AB)^T * (AB) = I.
Using the properties of transpose and matrix multiplication, we can simplify this equation.
(AB)^T * (AB) = B^T * A^T * A * B = B^T * I * B = B^T * B.
Since A and B are orthogonal matrices, A^T * A = I and B^T * B = I.
Therefore, (AB)^T * (AB) = I, which means the product of two orthogonal matrices is also an orthogonal matrix.
Hence, the correct answer is option C: Orthogonal matrix.
Two matrices are orthogonal if their product is the identity matrix.
Let's assume that we have two orthogonal matrices, A and B, and we want to find the product of these matrices.
Product of two matrices:
The product of two matrices is obtained by multiplying the corresponding entries of the matrices and adding them up.
If A is an m×n matrix and B is an n×p matrix, then the product of A and B is an m×p matrix.
Orthogonal matrices:
A matrix A is orthogonal if its transpose is equal to its inverse.
A matrix A is orthogonal if A^T * A = I, where I is the identity matrix.
If A and B are orthogonal matrices, then (AB)^T * (AB) = I.
Using the properties of transpose and matrix multiplication, we can simplify this equation.
(AB)^T * (AB) = B^T * A^T * A * B = B^T * I * B = B^T * B.
Since A and B are orthogonal matrices, A^T * A = I and B^T * B = I.
Therefore, (AB)^T * (AB) = I, which means the product of two orthogonal matrices is also an orthogonal matrix.
Hence, the correct answer is option C: Orthogonal matrix.
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