Describe geometrically the action of an orthogonal $3\times 3$ matrix with determinant -1.

Verify that the natural basis for $R^n$ is an orthonormal set.

Prove that an n x n matrix with complex entries is unitary if and only if the columns of A form an orthonormal set in

$$C^n.$$

Solve the following exercise about orthogonal vectors and orthogonal matrices.

(a) If A has three orthogonal columns each of length 4 , what is $A^{\mathrm{T}} A$ ?

(b) If A has three orthogonal columns of lengths $1,2,3$, what is $A^{\mathrm{T}} A$ ?

Determine whether the statement is true or false. If the columns of an $n\times n$ matrix form an orthogonal basis for $R^n$, then the matrix is an orthogonal matrix.

Prove the Theorem : The $n \times n$ matrix $A$ is orthogonal if and only if the columns (and rows) of $A$ form an orthonormal set of vectors in $R^n$.

For any $n \times m$ matrix A there exists an orthogonal $m \times m$ matrix S such that the columns of matrix AS are orthogonal.

Prove that if the points (x_1, y_1) and (x_2, y_2) lie on the same line as (x_1*, y_1*) and (x_2*, y_2*), then y_2* - y_1* / x_2* - x_1* = y_2 - y_1 / x_2 - x_1. Assume x_1 ≠ x_2 and x_1* ≠ x_2*.

For two invertible $n \times n$ matrices A and B, what is the relationship between adj(A), adj(B), and adj(AB)?

Mark each of the following True or False.

a. A square matrix is orthogonal if its column vectors are orthogonal.

b. Every orthogonal matrix has nullspace $\{0\}$.

c. If $A^T$ is orthogonal, then $A$ is orthogonal.

d. If $A$ is an $n \times n$ symmetric orthogonal matrix, then $A^2=I$.

e. If $d$ is an $n \times n$ symmetric matrix such that $A^2=I$, then $A$ is orthogonal.

f. If $A$ and $B$ are orthogonal $n \times n$ matrices, then $A B$ is orthogonal.

g. Every orthogonal linear transformation carries every unit vector into a unit vector.

h. Every linear transformation that carries each unit vector into a unit vector is orthogonal.

i. Every map of the plane into itself that is an isometry (that is, preserves distance between points) is given by an orthogonal linear transformation.

j. Every map of the plane into itself that is an isometry and that leaves the origin fixed is given by an orthogonal linear transformation.

Consider $A$ and $B$ to be orthogonal matrices of the same size. Prove that the product matrix $A B$ is also an orthogonal matrix.

Verify that the matrix $P$ in example discussed before is an orthogonal matrix.

Define an orthogonal matrix.

Let A be an $m \times n$ matrix with real entries. Prove that A = 0 if and only if trace $\left(A^{t} A\right)=0$.