RECORD. RECORD COLLABORATE: Discuss: Is the statement below correct? Try a 2x2 example.

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Presentation transcript:

RECORD

COLLABORATE: Discuss: Is the statement below correct? Try a 2x2 example.

Matrix operations : Transposition properties of matrix transposition

Example: Matrix equations

Powers of Matrices: Must A be a square matrix? k times

Partitioned Matrices: partition lines (horizontal, vertical) blocks of a partitioned matrix

Multiplication of Partitioned Matrices: block-wise multiplication The column partition of A needs “to match” the row partition of B: # of horizontal blocks in A = # of vertical blocks in B # of columns in each horizontal block in A = # of rows in the corresponding vertical block of B 1

Multiplication of Partitioned Matrices:

Block-Diagonal Matrices: a block-diagonal matrix: square blocks are on the diagonal zero blocks everywhere else all square matrices

Multiplication of Block-Diagonal Matrices:

Special Forms of Matrices: diagonal matrix symmetric matrix skew-symmetric matrix the identity matrix

Special Forms of Matrices: upper triangular lower triangular

Special Forms of Matrices: Reduced Row Echelon Form (rref) or “row reduced form” Row Echelon Form (ref): Leading 1’s. Leading 1’s are shifted to the right. All zero rows are on the bottom. Elements below leading 1’s are zero.

Systems of Linear Equations: Examples:

Systems of Linear Equations: the matrix notation Examples: With the augmented matrix [A|b]

Systems of Linear Equations: number of solutions 2-space: 3-space:

Equivalent Elementary Operations: Systems and Matrices

Gaussian Elimination: derived system back-substitution

Exercise: Solve the system by Gaussian elimination: derived system back-substitution

Exercise: Solve the system by Gaussian elimination: