Matrix Operations

Concept:
Determinants - Column Operations and Algebraic Identities.

• If two columns (or rows) of a determinant are proportional, the value of the determinant is .

Step 1: Apply the algebraic identity to the determinant columns.
Let's perform a column operation: replace Column 1 ( ) with . $ (x+y)^2 - (x-y)^2 = 4xy (10^5+10^{-5})^2 - (10^5-10^{-5})^2 = 4(10^5)(10^{-5}) = 4(1) = 4 (100^5+100^{-5})^2 - (100^5-100^{-5})^2 = 4(100^5)(100^{-5}) = 4(1) = 4 (6^{100}+6^{-100})^2 - (6^{100}-6^{-100})^2 = 4(6^{100})(6^{-100}) = 4(1) = 4 C_1 4 C_3 C_1 C_3$ are proportional, the value of the determinant is zero.

Concept:
Matrices - Matrix Multiplication and Trigonometric Identities.
Recall the fundamental identity: .
Step 1: Find the transpose of matrix A.
Given the matrix: $ A^T AA^T (1,1) (1)(1) + (\tan x)(\tan x) = 1 + \tan^2 x (1,2) (1)(-\tan x) + (\tan x)(1) = -\tan x + \tan x = 0 (2,1) (-\tan x)(1) + (1)(\tan x) = -\tan x + \tan x = 0 (2,2) (-\tan x)(-\tan x) + (1)(1) = \tan^2 x + 1 1 + \tan^2 x = \sec^2 x $

Concept:
Determinants - Row/Column Operations and Factorization.
Step 1: Observe row/column sums to find a common factor.
Notice that the sum of the elements in each row is the same:

• Row 1 sum:
• Row 2 sum:
• Row 3 sum:

Apply the column operation: . $ C_1 (x+19) C_1 R_2 \rightarrow R_2 - R_1 R_3 \rightarrow R_3 - R_1 C_1 \Delta = (x-2)^2(ax+b) a = 1 b = 19$.

Concept:
Matrices - Properties of Symmetric and Skew-Symmetric Matrices.

• A matrix is symmetric if .
• A matrix is skew-symmetric if .
• Transpose property: and .

Step 1: Check the symmetry of .
Let . Take the transpose of : $ P^T = P A + A^T A - A^T Q = A - A^T Q Q^T = -Q A - A^T$ is skew-symmetric.