Transcription of Selected Problems — Matrix Algebra Math 2300
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Selected Problems — Matrix Algebra Math 2300
Selected Problems Matrix AlgebraMath that ifAis nonsingular thenATis nonsingular and(AT) 1= (A 1) :Lets put into words what are we asked to show in this problem. First, we mustshow that if a Matrix is invertible, then so is its transpose. We must also show that theinverse of the transpose is the same as the transpose of the inverse. In other words, if wethink of inverting and transposing as processes we may perform on square matrices, then forinvertible matrices, these two processes performed in either order yield the same :LetAbe nonsingular. By definition, there existsA 1such thatA 1A=AA 1= we note thatIT=I(verify this).
matrix and a skew-symmetric matrix that add to give 2A, the matrix A times the scalar 2. We fix the problem by multiplying both sides of (4) by 1/2. 1 2 [(A+AT)+(A−AT)] = 1 2 (2A) =⇒ 1 2 (A+AT)+ 1 2 (A−AT) = A since scalar multiplication distributes over matrix addition. Finally, we note that multiplying a symmetric matrix by a scalar ...
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