Section 5.1: Matrices

1. Describe how matrices, vectors, and scalars differ in how they correspond to an array (“organized box of numbers”).
2. Recognize the notation (in writing) \(\underline{\underline{A}}\), \(\underline{b}\), and \(c\) and (in typeset notes) \(\mathbf{A}\), \(\mathbf{b}\), and \(c\) for a matrix, vector, and scalar.
3. Recognize that we specify the size of a matrix \(\mathbf{A}\) as \(m \times n\), where \(m\) is the number of rows and \(n\) is the number of columns.
4. Recognize the notation \(a_{ij}\) or \((\mathbf{A})_{ij}\) for the matrix entry in the \(i\)-th row and \(j\)-th column.
5. Given a matrix \(\mathbf{A}\), compute its transpose \(\mathbf{A}^T\).
6. State what a matrix equality \(\mathbf{A} = \mathbf{B}\) means in terms of the elements of the matrices.
7. Identify the design matrix \(\mathbf{X}\) and response vector \(\mathbf{Y}\) for a simple linear regression model.

Section 5.2: Matrix Addition and Subtraction

1. Given two matrices \(\mathbf{A}\) and \(\mathbf{B}\), perform the matrix operations \(\mathbf{A} + \mathbf{B}\) and \(\mathbf{A} - \mathbf{B}\).

Section 5.3: Matrix Multiplication

1. Given a matrix \(\mathbf{A}\) and a scalar \(k\), compute the matrix-scalar product \(k\mathbf{A}\).
2. Given a matrix \(\mathbf{A}\) and a vector \(\mathbf{b}\), compute the matrix-vector product \(\mathbf{A}\mathbf{b}\) both using (1) the direct definition of a matrix-vector product and (2) the row-column rule.
3. Identify the conditions on the dimension of a matrix \(\mathbf{A}\) and a vector \(\mathbf{b}\) for their matrix-vector product to be well-defined.
4. Given two matrices \(\mathbf{A}\) and \(\mathbf{B}\), compute the matrix-matrix product \(\mathbf{A}\mathbf{B}\) both using (1) the direct definition of a matrix-matrix product and (2) the row-column rule.
5. Identify the conditions on the dimension of the matrices \(\mathbf{A}\) and \(\mathbf{B}\) for their matrix-matrix product to be well-defined.
6. Explain what it means to say that matrix-matrix multiplication is not commutative.
7. Compute and simplify matrix-matrix and matrix-vector products relevant to simple linear regression, for example \(\mathbf{Y}^{T}\mathbf{Y}\), \(\mathbf{X}^{T}\mathbf{X}\), and \(\mathbf{X}^{T}\mathbf{Y}\).

Section 5.4: Special Types of Matrices

1. State the \(n \times n\) identity matrix \(\mathbf{I}_{n}\).
2. Explain why the identity matrix is so-named.

Section 5.6: Inverse of a Matrix

1. Define the multiplicative inverse of a scalar \(a \neq 0\).
2. Define the multiplicative inverse of an invertible matrix \(\mathbf{A}\).
3. Explain why in matrix arithmetic, we cannot write \(\mathbf{A}/\mathbf{B}\).
4. Compute the inverse of a \(2 \times 2\) matrix.
5. Verify that a given matrix \(\mathbf{A}^{-1}\) is the inverse of another matrix \(\mathbf{A}\).
6. Define invertible / non-invertible and singular / non-singular in the context of matrix inverses.
7. Compute the inverse of the matrix \(\mathbf{X}^{T} \mathbf{X}\) from a simple linear regression.

R

1. Use the function matrix from Base R to construct matrices and vectors.
2. Use the function string2matrix from MUsaic to construct matrices and vectors.
3. Recognize MATLAB syntax for constructing a matrix or vector.
4. Perform matrix arithmetic using R, including matrix sums and differences, matrix-vector products, matrix-matrix products.
5. Compute the inverse of a matrix using R.