Matrix Addition: Sums matrices by adding corresponding elements. Dimensions MUST match!
Ex: `[[1,2],[3,4]] + [[5,6],[7,8]] = [[6,8],[10,12]]`
Pro-Tip: If dimensions differ, addition is undefined! Crucial for compatibility in systems.
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**Identity Matrix (I):** Acts like the number '1' for matrices.
Ex: Multiplying any matrix A by I gives A back: `A*I = I*A = A`.
Pro-Tip: It's square, with 1s on the main diagonal & 0s elsewhere. Essential for inverse ops!
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**Determinant:** A scalar value from a square matrix. Reveals invertibility & volume scaling.
Ex: For `[[a,b],[c,d]]`, `det(A) = ad - bc`.
Pro-Tip: If `det(A) = 0`, the matrix is singular & its columns are linearly dependent!
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**Matrix Transpose (Aᵀ):** Flips a matrix over its diagonal, turning its rows into columns & vice versa.
Ex: If `A = [[1,2],[3,4]]`, then `Aᵀ = [[1,3],[2,4]]`.
Pro-Tip: For product `(AB)ᵀ = BᵀAᵀ` (note the flipped order!)
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**Elementary Row Ops:** Operations to transform a matrix (swap rows, scale rows, add rows).
Ex: `R1 <-> R2` (swap row 1 & row 2).
Pro-Tip: Key for solving systems of equations or finding matrix inverses!
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**Matrix Multiplication:** Creates a new matrix where each entry is the dot product of a row from the 1st and a col from the 2nd.
Ex: `(AB)_ij = sum(A_ik * B_kj)`
Pro-Tip: The # of columns in the first matrix MUST equal the # of rows in the second!
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**Matrix Transpose:** Switches a matrix's rows & columns.
Ex: If `A = [[1,2],[3,4]]`, then `A^T = [[1,3],[2,4]]`.
Pro-Tip: Dimensions flip! An `m x n` matrix becomes `n x m`.
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**Determinant:** A scalar from a square matrix, indicating if it's invertible.
Ex: For `A = [[a,b],[c,d]]`, `det(A) = ad - bc`.
Pro-Tip: If `det(A) = 0`, the matrix is singular (no inverse exists!).
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The **Identity Matrix (I)** acts like the number '1' for matrix multiplication.
Ex: `A * I = A`. For 2x2: `I = [[1,0],[0,1]]`.
Pro-Tip: It's always square & has 1s on the main diagonal, 0s everywhere else.
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Alright, future engineers!
**Identity Matrix (I):** A square matrix with 1s on the main diagonal & 0s elsewhere.
Ex: `I_2 = [[1,0],[0,1]]`
Pro-Tip: It's the '1' of matrix multiplication! `A*I = I*A = A`.
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