Matrices
- What are symmetric and skew-symmetric matrices, and how is any square matrix written as their sum? In one line A square matrix is symmetric if A^T=A and skew-symmetric if A^T=-A ; every square matrix equals 1/2(A+A^T)+1/2(A-A^T) , a symmet... 1
- How do I find the inverse of a 3×3 matrix using the adjoint method? In one line If |A|≠0 , then A⁻¹=1/|A| adj A , where adj A is the transpose of the matrix of cofactors. Explained simply The inverse of a num... 1
- What are Hermitian and skew-Hermitian matrices and what properties do they have? In one line A complex square matrix is Hermitian if it equals its conjugate transpose ( A^θ=A ) and skew-Hermitian if A^θ=-A ; they are the... 1
- What are orthogonal and unitary matrices and how do I check whether a matrix is one? In one line A real matrix is orthogonal if AA^T=I (so A⁻¹=A^T ); a complex matrix is unitary if AA^θ=I (so A⁻¹=A^θ ). Explained simply Rotat... 1
- What is the rank of a matrix and how do elementary transformations help find it? In one line The rank of a matrix is the order of its largest non-zero minor, which equals the number of linearly independent rows; elementar... 1
- How do I find the rank of a matrix by reducing it to echelon form? In one line Use row operations to make the matrix upper-triangular in a staircase pattern (echelon form); the rank is the number of non-zero... 1
- What is the normal form of a matrix and how do I reduce a matrix to it to find rank? In one line The normal form of a matrix is beginbmatrixI_r&0\\0&0endbmatrix , obtained with both row and column operations; the size r of th... 1
- How do I find non-singular matrices P and Q such that PAQ is in normal form? In one line Write A=I A I ; every row operation on A is also done on the left identity (building P ) and every column operation on the right... 1
- When does a homogeneous system AX = 0 have non-trivial solutions, and how do I find them? In one line AX=0 always has the trivial solution X=0 ; it has non-trivial (infinitely many) solutions exactly when ρ(A)<n , the number of un... 1
- How do I test a non-homogeneous system AX = B for consistency and find its solutions? In one line Compare the rank of A with the rank of the augmented matrix [A | B] : unequal ranks mean no solution; equal ranks =n mean a uniq... 1
- How is the inverse of a matrix used to encode and decode messages in coding theory? In one line A message is turned into number columns, scrambled by multiplying with an invertible matrix A , and unscrambled by multiplying w... 1
- Investigate for what values of \(\lambda\) and \(\mu\) the equations \(x+y+z=6,\; x+2y+3z=10,\; x+2y+\lambda z=\mu\) will have (i) No Solution (ii) Unique Solution (iii) Infinitely Many Solutions. Augmented matrix: [Amid B]=[beginarrayccc|c1&1&1&6\\1&2&3&10\\1&2&λ&μendarray] R₂→ R₂-R₁, R₃→ R₃-R₁ : [beginarrayccc|c1&1&1&6\\0&1&2&4\\0&1&... 1
- Find non-singular matrices P & Q such that PAQ is in normal form & hence find rank of the matrix A and obtain \(A^{-1}\). \(A=\begin{bmatrix}1&2&-2\\-1&3&0\\0&-2&1\end{bmatrix}\) Write A=I₃ A I₃ . R₂→ R₂+R₁ : beginbmatrix1&2&-2\\0&5&-2\\0&-2&1endbmatrix=beginbmatrix1&0&0\\1&1&0\\0&0&1endbmatrixA I₃ C₂→ C₂-2C₁,\ C₃→ C₃... 1
- Reduce the following matrix to normal form and find the rank of matrix \(A=\begin{bmatrix}1&1&1\\1&-1&-1\\3&1&1\end{bmatrix}\). R₂→ R₂-R₁,\ R₃→ R₃-3R₁ : beginbmatrix1&1&1\\0&-2&-2\\0&-2&-2endbmatrix R₃→ R₃-R₂ : beginbmatrix1&1&1\\0&-2&-2\\0&0&0endbmatrix C₂→ C₂-C₁,\ C... 1
- For the matrix \(A=[a_{ij}]_{3\times3}\), where \(a_{ij}=i+j\), find the rank. A=beginbmatrix2&3&4\\3&4&5\\4&5&6endbmatrix R₂→ R₂-R₁,\ R₃→ R₃-R₁ : beginbmatrix2&3&4\\1&1&1\\2&2&2endbmatrix ; R₃→ R₃-2R₂ : beginbmatrix2&3... 1
- Express the matrix \(A=\begin{bmatrix}2+3i&2&3i\\-2i&0&1+2i\\4&2+5i&-i\end{bmatrix}\) as the sum of a Hermitian and a skew-Hermitian matrix. A=P+Q with P=1/2(A+A^θ) (Hermitian) and Q=1/2(A-A^θ) (skew-Hermitian), where A^θ=(bar A)^T . A^θ=beginbmatrix2-3i&2i&4\\2&0&2-5i\\-3i&1-2i&i... 1
- Find the rank of the matrix by reducing it to normal form \(A=\begin{bmatrix}1&-1&-2&-3\\4&1&0&2\\0&3&1&4\\0&1&0&2\end{bmatrix}\). C₂→ C₂+C₁,\ C₃→ C₃+2C₁,\ C₄→ C₄+3C₁ : beginbmatrix1&0&0&0\\4&5&8&14\\0&3&1&4\\0&1&0&2endbmatrix R₂→ R₂-4R₁ , then R₂rightarrow R₄ : beginbma... 1
- Test the consistency and hence solve the following system of linear equations, \(-2y+3t=2,\ 2x+y+z+t=-4,\ 4x-3y+z+7t=8\). [As printed; the first equation is very likely \(x-2y+3t=2\), with the \(x\) dropped in print.] The first equation is printed as -2y+3t=2 , with no x-term visible. The standard textbook problem is x-2y+3t=2 , so that is solved first; th... 1
- Investigate for what values of \(\lambda\) and \(\mu\) the system of equations \(x+y+z=6,\; x+2y+3z=10,\; x+2y+\lambda z=\mu\) will have (i) No Solution, ii) Unique Solution iii) Many Solutions. Augmented matrix: [Amid B]=[beginarrayccc|c1&1&1&6\\1&2&3&10\\1&2&λ&μendarray] R₂→ R₂-R₁, R₃→ R₃-R₁ : [beginarrayccc|c1&1&1&6\\0&1&2&4\\0&1&... 1
- Show that the matrix \(A=\frac{1}{\sqrt3}\begin{bmatrix}1 & 1+i\\ 1-i & -1\end{bmatrix}\) is unitary and hence find \(A^{-1}\). Given: A=1/sqrt3beginbmatrix1 & 1+i\\ 1-i & -1endbmatrix . A is unitary if AA^θ=I , where A^θ=(bar A)^T . Step 1: bar A=1/sqrt3beginbmatrix1... 1
- Find nonsingular matrices P & Q such that PAQ is in normal form and hence find rank of the matrix \(A\) for the following matrix \(A=\begin{bmatrix}3&-3&4\\2&-3&4\\0&-1&1\end{bmatrix}\). Write A=I₃ A I₃ . Row operations act on A and the pre-factor; column operations act on A and the post-factor. R₁→ R₁-R₂ : beginbmatrix1&0&0\... 1
- Find \(a,b,c\) & \(A^{-1}\) if \(A=\frac19\begin{bmatrix}8&-4&a\\1&4&b\\4&7&c\end{bmatrix}\) is orthogonal. A is orthogonal ⇒ its columns are orthonormal. Check: C₁· C₁=64+1+16/81=1,\ C₂· C₂=16+16+49/81=1,\ C₁· C₂=-32+4+28/81=0 . ✓ C₃⊥ C₁:\ 8a+b+4c... 1
- Investigate for what values of \(\lambda\) and \(\mu\) the system of linear equations \(x+y+z=6,\; x+2y+3z=10,\; x+2y+\lambda z=\mu\) will have (i) No solution, (ii) Unique solution, (iii) Infinite number of solutions. Augmented matrix: [Amid B]=[beginarrayccc|c1&1&1&6\\1&2&3&10\\1&2&λ&μendarray] R₂→ R₂-R₁, R₃→ R₃-R₁ : [beginarrayccc|c1&1&1&6\\0&1&2&4\\0&1&... 1