# If A,B are symmetric matrices, then prove that (BA^{-1})^T(A^{-1}B^T)^{-1} = I

Question
Matrices
If A,B are symmetric matrices, then prove
that $$(BA^{-1})^T(A^{-1}B^T)^{-1} = I$$

2020-12-31
Step 1
Given that, A,B are symmetric matrices.
Prove that $$(BA^{-1})^T(A^{-1}B^T)^{-1} = I$$
Step 2
Consider the LHS,
$$(BA^{-1})^T(A^{-1}B^T)^{-1} = (A^{-1})^T(B)^T(B^T)^{-1}(A^{-1})^{-1}$$
$$=(A^{-1})^T IA \ \ \ \ \left[ \because (B^T)(B^T)^{-1}=I \text{ and } (A^{-1})^{-1}=A \right]$$
$$=(A^{-1})^T A^T \ \ \ \ \left[ \because A=A^T \right]$$
$$=(A^{T})^{-1} A^T \ \ \ \ \left[ \because (A^T)^{-1}=(A^{-1})^T \right]$$
$$=I \ \ \ \ \left[ \because (A^T)(A^T)^{-1}=I \right] \(=RHS$$
Hence , the required is obtained.

### Relevant Questions

In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.
Let A and B be $$n \times n$$ matrices. Recall that the trace of A , written tr(A),equal
$$\sum_{i=1}^nA_{ii}$$
Prove that tr(AB)=tr(BA) and $$tr(A)=tr(A^t)$$
Prove that if A and B are n x n matrices, then tr(AB) = tr(BA).
Determine whether the given set S is a subspace of the vector space V.
A. V=$$P_5$$, and S is the subset of $$P_5$$ consisting of those polynomials satisfying p(1)>p(0).
B. $$V=R_3$$, and S is the set of vectors $$(x_1,x_2,x_3)$$ in V satisfying $$x_1-6x_2+x_3=5$$.
C. $$V=R^n$$, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.
D. V=$$C^2(I)$$, and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.
E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.
F. V=$$P_n$$, and S is the subset of $$P_n$$ consisting of those polynomials satisfying p(0)=0.
G. $$V=M_n(R)$$, and S is the subset of all symmetric matrices
Prove: If A and B are $$n \times n$$ diagonal matrices, then
AB = BA.

For the following statement, either prove that they are true or provide a counterexample:
Let a, b, c, $$\displaystyle{m}\in{Z}$$ such that m > 1. If $$\displaystyle{a}{c}\equiv{b}{c}{\left(\mod\right)},{t}{h}{e}{n}\ {a}\equiv{b}{\left(\mod{m}\right)}$$

If the product D=ABC of three square matrices is invertible , then A must be invertible (so are B and C). Find a formula for $$A^{-1} (i.e. A^{-1}=\dotsb) that involves only the matrices \(A, B B^{-1} , C, C^{-1} , D \text{ and/or } D^{-1}$$
Let W be the vector space of $$3 \times 3$$ symmetric matrices , $$A \in W$$ Then , which of the following is true ?
a) $$A^T=1$$
b) $$dimW=6$$
c) $$A^{-1}=A$$
d) $$A^{-1}=A^T$$
Determine whether each of the following statements is true or false, and explain why.If A and B are square matrices of the same size, then AB = BA
Prove directly from the definition of congruence modulo n that if a,c, and n are integers,n >1, and $$\displaystyle{a}\equiv{c}{\left({b}\text{mod}{n}\right)}$$, then $$\displaystyle{a}^{{{3}}}\equiv{c}^{{{3}}}{\left({b}\text{mod}{n}\right)}$$.