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Consider the following four matrices.
\begin{align*}
A=\begin{bmatrix}
4 & 6 \\
-2 & -3
\end{bmatrix}\quad B=\begin{bmatrix}
4 \\
2 \\
8
\end{bmatrix} \quad C=\begin{bmatrix}
5 & 0 \\
2 & 5 \\
3 & 1
\end{bmatrix}\quad D=\begin{bmatrix}
3 & -2 & 5
\end{bmatrix}
\end{align*}
Which one of the following computations is defined?
\(B\)
\(\text{Option A: \( A+B\) is not defined (eliminate)}\)
\(\text{Option B: \(C^{\text{T}}BD\) is defined (see below)}\)
\(C^{\text{T}}=(2 \times 3)\ \text{matrix. }\ C^{\text{T}} B=(2 \times 3)(3 \times 1)=(2 \times 1)\)
\(C^{\text{T}} B D=(2 \times 1)(1 \times 3)=(2 \times 3)\)
\(\text{Option C: \(B D=(3 \times 3)\) matrix, \(C+B D\) is not defined (eliminate)}\)
\(\text{Option D: \(DA=(1 \times 3)(2 \times 2) \Rightarrow\) not defined (eliminate)}\)
\(\Rightarrow B\)
Matrix \(J\) is a row matrix of order \(1 \times n\).
Matrix \(K\) is a column matrix of order \(n \times 1\).
Matrix \(J^T\) is the transpose of Matrix \(J\).
Matrix \(K^T\) is the transpose of Matrix \(K\).
Consider the following matrix products where \(n\) is a whole number greater than or equal to 2:
How many of the above matrix products are defined?
\(D\)
\(\text{Note:}\ J^{T}\ \text{is order}\ (n \times 1), \ K^{T}\ \text{is order}\ (1 \times n) \)
\(J^2: (1 \times n) \times (1 \times n) \Rightarrow \text{Undefined}\)
\(JK: (1 \times n) \times (n \times 1) \Rightarrow \text{Defined}\)
\(KJ: (n \times 1) \times (1 \times n) \Rightarrow \text{Defined}\)
\(J^T K^T: (n \times 1) \times (1 \times n) \Rightarrow \text{Defined}\)
\(K^T J^T: (1 \times n) \times (n \times 1) \Rightarrow \text{Defined}\)
\(\Rightarrow D\)
Elena imports three brands of olive oil: Carmani (`C`) Linelli (`L`) and Ohana (`O`).
The number of 1 litre bottles of these oils sold in January 2021 is shown in matrix `J` below.
`J = {:[(2800),(1700),(2400)]:} {:(C),(L),(O):}`
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a. `3 xx 1`
b. `text{All brands} \ ↑ 5%`
`:. k = 1.05`
`A` is a 7 × 7 matrix.
`B` is a 10 × 7 matrix.
Which one of the following matrix expressions is defined?
`E`
`text{Consider each option}`
`A : \ AB-2B -> text{not defined (AB not defined)}`
`B : \ A(BA)^-1`
`BA -> (10 xx 7) (7 xx 7) -> (10 xx 7)`
`( BA)^-1 \ text{is not defined (inverse matrices only apply to square matrices)}`
`C : \ AB^2 -> text{not defined} \ (B^2 \ text{not defined – only square matrices can be raised to a power)}`
`D : \ A^2 – BA`
`A^2 -> (7 xx 7), \ BA -> (10 xx 7)`
`qquad \ A^2 – BA \ -> text{not defined (only matrices of the same order can be subtracted)}`
`E: \ A(B^T)`
`A -> (7 xx 7), \ B^T -> (7 xx 10)`
`A (B^T) \ text{is defined as Columns in} \ A = \ text{rows in} \ B^T`
`=> E`
Matrix `W` is a `3 xx 2` matrix.
Matrix `Q` is a matrix such that `Q xx W = W`.
Matrix `Q` could be
| A. | `\ [1]` | B. | `\ [(1,0),(0,1)]` | C. | `\ [(1,1),(1,1),(1,1)]` |
| D. | `\ [(1,0,0),(0,1,0),(0,0,1)]` | E. | `\ [(1,1,1),(1,1,1),(1,1,1)]` |
`D`
`W = (3 xx 2)`
`[(1,0,0),(0,1,0),(0,0,1)][(e_11,e_12),(e_21,e_22),(e_31,e_32)] = [(e_11,e_12),(e_21,e_22),(e_31,e_32)]`
`=>\ D`
Consider the following four matrix expressions.
`[(8), (12)]+[(4), (2)] qquad qquad qquad qquad qquad qquad quad [(8), (12)]+[(4, 0),(0, 2)]`
`[(8, 0),(12, 0)] + [(4), (2)] qquad qquad qquad qquad qquad \ [(8, 0),(12, 0)] + [(4, 0),(0, 2)]`
How many of these four matrix expressions are defined?
`C`
`[(8), (12)]+[(4), (2)] = [(12), (4)]`
`[(8), (12)]+[(4, 0),(0, 2)] = text(not defined)`
`[(8, 0),(12, 0)] + [(4), (2)] = text(not defined)`
`[(8, 0),(12, 0)] + [(4, 0),(0, 2)] = [(12, 0), (12, 2)]`
`=> C`
Matrix `A` is a `1 xx 3` matrix.
Matrix `B` is a `3 xx 1` matrix.
Which one of the following matrix expressions involving `A` and `B` is defined?
A. `A + 1/3 B`
B. `2B xx 3A`
C. `A^2 B`
D. `B^-1`
E. `B - A`
`B`
`text(Consider)\ B,`
`underset (3 xx 1) (2B) xx underset (1 xx 3) (3A)`
`text(S) text(ince the number of columns in)`
`text(matrix)\ 2B\ text(equals the number of)`
`text(rows in matrix)\ 3A,\ text(their product)`
`text(is defined.)`
`=> B`
`m` and `n` are positive whole numbers.
Matrix `P` is of order `m xx n.`
Matrix `Q` is of order `n xx m.`
The matrix products `PQ` and `QP` are both defined
A. for no values of `m` and `n`
B. when `m` is equal to `n` only
C. when `m` is greater than `n` only
D. when `m` is less than `n` only
E. for all values of `m` and `n.`
`E`
`underset (m xx n) P xx underset (n xx m) Q = underset (m xx m)(PQ)`
`:. PQ\ text(is defined for all values)`
`text(of)\ m and n.`
`text(Similarly,)`
`underset (n xx m)Q xx underset (m xx n) P = underset (n xx n) (QP)`
`=> E`
The order of the matrix `[(2, 2), (2, 2), (2, 2)]` is
A. `2 xx 2`
B. `2 xx 3`
C. `3 xx 2`
D. `4`
E. `6`
`C`
`=> C`
`P, Q, R` and `S` are matrices such that the matrix product `P = QRS` is defined.
Matrix `Q` and matrix `S` are square, non-zero matrices for which `Q + S` is not defined.
Which one of the following matrix expressions is defined?
A. `R - S`
B. `Q + R`
C. `P^2`
D. `R^-1`
E. `P × S`
`E`
`text(If)\ Q(n xx n)\ text(and)\ S(m xx m)\ text(are square)`
`text(matrices and)\ Q + S\ text(is not defined,)`
`m != n.`
`text(S)text(ince)\ QRS\ text(is defined, we can deduce)`
`text(that)\ R\ text(is)\ n xx m,\ text(and)\ P\ text(is)\ n xx m.`
`text(Consider)\ E,`
`P xx S\ text(is defined because the number)`
`text(of columns in)\ P(m)\ text(equals the number)`
`text(of rows in)\ S(m).`
`text(All other options can be shown to not)`
`text(be defined.)`
`rArr E`
Matrix `A` has three rows and two columns.
Matrix `B` has four rows and three columns.
Matrix `C = B × A` has
A. two rows and three columns.
B. three rows and two columns.
C. three rows and three columns.
D. four rows and two columns.
E. four rows and three columns.
`D`
| `B` | `xx` | `A` | `=` | `C` |
| `4 xx 3` | `3 xx 2` | `4 xx 2` |
`rArr D`
If `A = [(1,3), (6,4), (0,0)]` and the matrix product `XA = [(4,1),(1,4), (3,5)],` then the order matrix `X` is
A. `(2 xx 2)`
B. `(2 xx 3)`
C. `(3 xx 1)`
D. `(3 xx 2)`
E. `(3 xx 3)`
`E`
| `X` | `xx` | `A` | `=` | `XA` |
| `a xx b` | `3 xx 2` | `3 xx 2` |
`b = 3\ \ text{(columns in}\ X = text(rows in)\ A)`
`a = 3\ \ text{(rows in}\ X = text(rows in)\ XA)`
`:. XA\ text(is order)\ 3 xx 3.`
`rArr E`
Let `A = [(-2), (0)], B = [(0, 9)]` and `C = [2]`
Using these matrices, the matrix product that is not defined is
A. `AB`
B. `AC`
C. `BA`
D. `BC`
E. `CB`
`D`
`text(Consider)\ D,`
| `B` | `xx` | `C` |
| `1 xx 2` | `1 xx 1` |
`text(Columns in)\ B !=\ text(Rows in)\ C`
`:. BC\ text(is undefined.)`
`rArr D`
Matrix `M` is a `3 xx 4` matrix.
Matrix `P` has five rows.
`N` is another matrix.
If the matrix product
`M (NP) = [(4, 1, 7, 2), (0, 9, 7, 4), (4, 3, 3, 1)],`
then the order of matrix `N` is
A. `3 xx 5`
B. `5 xx 3`
C. `4 xx 5`
D. `5 xx 4`
E. `5 xx 5`
`C`
`M xx (NP) = [(4,1,7,2),(0,9,7,4),(4,3,3,1)]`
`text{The order of matrices (above) is}`
`(3 xx 4) xx ((a xx b)(5 xx c)) = (3 xx 4)`
`a = 4, b = 5, c = 4`
`:. N\ text{is a (4 × 5) matrix.}`
`=> C`
Matrix `A` is a 3 x 4 matrix.
Matrix `B` is a 3 x 3 matrix.
Which one of the following matrix expressions is defined?
A. `BA^2`
B. `BA - 2A`
C. `A + 2B`
D. `B^2 - AB`
E. `A^-1`
`B`
`text(Consider)\ B,`
| `B` | `xx` | `A` | `=` | `BA` |
| `3 xx 3` | `3 xx 4` | `3 xx 4` |
| `:. BA` | `-` | `2Aqquadtext(is defined)` |
| `3 xx 4` | `3 xx 4` |
`text(All other options can be shown)`
`text(to produce undefined matrices.)`
`=> B`