- A matrix A over R is called symmetric if At = A and skew-symmetric if At = −
- A matrix A is said to be orthogonal if AAt = AtA =

Examples

- Let

and

Then A is a symmetric matrix and B is a skew-symmetric matrix. - Let

Then A is an orthogonal matrix. - Let A = [aij ] be an n×n matrix with

Then An = 0 and Aℓ 6= 0 for 1 ≤ ℓ ≤n − 1. The matrices A for which a positive integer k exists such that Ak = 0 are called nilpotent The least positive integer k for which Ak = 0 is called the order of nilpotency. - Let

Then A2 = The matrices that satisfy the condition that A2 = A are called idempotent matrices.- Show that for any square matrix A, S = 2 (A + At) is symmetric, T = 1/2(A+A
^{t}) skew-symmetric, and A = S + - Show that the product of two lower triangular matrices is a lower triangular matrix. A similar statement holds for upper triangular matrices.
- Let A and B be symmetric matrices. Show that AB is symmetric if and only if AB =
- Show that the diagonal entries of a skew-symmetric matrix are zero.
- Let A,B be skew-symmetric matrices with AB = Is the matrix AB symmetric or skew-symmetric?
- Let A be a symmetric matrix of order n with A2 = 0. Is it necessarily true that A = 0?
- Let A be a nilpotent matrix. Show that there exists a matrix B such that B(I + A) = I = (I + A)

**Submatrix of a Matrix**For example, if

a few submatrices of A are

But the matrices

**Miscellaneous Exercises**

Let

Geometrically interpret y = Ax

**SOME MORE SPECIAL MATRICES**- Compose the two transformations to express x1, x2 in terms of z1, z2.
- If xt = [x1, x2], yt = [y1, y2] and zt = [z1, z2] then find matrices A,B and C such that x = Ay, y = Bz and x = Cz.
- Is C = AB?
- For a square matrix A of order n, we define trace of A, denoted by tr (A) as tr (A) = a11 + a22 + · · · ann.

Then for two square matrices, A and B of the same order, show the following:

(a) tr (A + B) = tr (A) + tr (B).

(b) tr (AB) = tr (BA). - Show that, there do not exist matrices A and B such that AB − BA = cIn for any c 6= 0.
- Let A and B be two m × n matrices and let x be an n × 1 column vector.

(a) Prove that if Ax = 0 for all x, then A is the zero matrix.

(b) Prove that if Ax = Bx for all x, then A = B. - Let A be an n × n matrix such that AB = BA for all n × n matrices B. Show that A = αI for some α ∈
- Let

Show that there exist infinitely many matrices B such that BA = I2. Also, show that there does not exist any matrix C such that AC = I3.

**Block Matrices**

Let A be an n × m matrix and B be an m × p matrix. Suppose r < m. Then, we can decompose the matrices A and B as A = [P Q] and

where P has order n × r and H has order r × p. That is, the matrices P and Q are submatrices of A and P consists of the first r columns of A and Q consists of the last m − r columns of A. Similarly, H and K are submatrices of B and H consists of the first r rows of B and K consists of the last m− r rows of B. We now prove the following important theorem.

__Theorem__

Proof. First note that the matrices PH and QK are each of order n × p. The matrix products PH and QK are valid as the order of the matrices P,H,Q and K are respectively, n ×r, r ×p, n ×(m− r) and (m−r)×p. Let P = [Pij ], Q = [Qij ], H = [Hij ], and K = [kij ]. Then, for 1 ≤ i ≤ n and 1 ≤ j ≤ p, we have

Theorem@ is very useful due to the following reasons:

- The order of the matrices P,Q,H and K are smaller than that of A or B.
- It may be possible to block the matrix in such a way that a few blocks are either identity matrices or zero matrices. In this case, it may be easy to handle the matrix product using the block form.
- Or when we want to prove results using induction, then we may assume the result for r × r submatrices and then look for (r + 1) × (r + 1) submatrices, etc.

For example, if

- Show that for any square matrix A, S = 2 (A + At) is symmetric, T = 1/2(A+A