Yaadannoo boqonnaa fi gaaffii shaakalaa

Boqonnaa 4 Herrega kutaa 11: Ditarminaantota fi Amaloottan Isaanii

Wanti fuula kana irra jiru hundi boqonnaa 4 kitaaba Herrega kutaa 11 tiif: wanta boqonnaan haguugu, yaadannoo gabaabaa, gaaffii shaakalaa deebii isaanii waliin, fi viidiyoo barnootaa bakka jirutti.

4
Kaardii: 10
Gaaffii shaakalaa boqonnaa 4: 207–250
Fuula kitaabaa boqonnaa 4

Boqonnaan 4 maal haguuga

1.Determinants of Matrices of Order
2.Minors and Cofactors of Elements of Matrices
3.Properties of Determinants
4.Solutions of Systems of Linear Equations Using Cramer's Rule
5.Applications

Yaadannoo gabaabaa boqonnaa 4

Kaardiiwwan fuula Herrega kutaa 11 irratti boqonnaa kana jala galmaa'an. Tokko tokkoon bani.

Bal'ina Rog-sadii Diriiroo xy KeessattiFoormulaa

\text{Area} = \frac{1}{2} \left| \det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} \right|

\text{Area} = \frac{1}{2} \left| \det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} \right|

$\text{Area}$: Bal'ina rog-sadii
$x_1, y_1$: Qindii qabxii tokkoffaa
$x_2, y_2$: Qindii qabxii lammaffaa
$x_3, y_3$: Qindii qabxii sadaffaa

Yoom akka itti fayyadamtu

Qindiin qabxiilee fiixee sadanii yoo kenname bal'ina rog-sadii shallaguuf itti fayyadamaa.

Of eeggadhu

Determinantiin lakkoofsa negaatiivii ta'uu danda'a. Bal'inni poozatiivii akka ta'uuf yeroo hunda gatii gonkaa fudhadhaa.

Grade 11 Mathematics, fuula 207-250 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Qabxiilee Sarar-tokkee MirkaneessuuFoormulaa

\det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} = 0

\det \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix} = 0

$x_1, y_1$: Qindii qabxii A
$x_2, y_2$: Qindii qabxii B
$x_3, y_3$: Qindii qabxii C

Yoom akka itti fayyadamtu

Qabxiileen sadan sarara qajeelaa tokko irra jiraachuu isaanii qoruuf itti fayyadamaa.

Of eeggadhu

Kana bal'ina shallaguu wajjin walitti hin makin. Sarar-tokkee ta'uuf determinantiin sirriitti zeeroo ta'uu qaba.

Grade 11 Mathematics, fuula 207-250 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Hima Sarara Qabxii Lama Keessa DarbuFoormulaa

\det \begin{pmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{pmatrix} = 0

\det \begin{pmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{pmatrix} = 0

$x, y$: Jijjiiramoota qabxii sararichaa kamiyyuu bakka bu'an
$x_1, y_1$: Qindii qabxii beekamaa tokkoffaa
$x_2, y_2$: Qindii qabxii beekamaa lammaffaa

Yoom akka itti fayyadamtu

Hima sarara qajeelaa qabxiilee adda addaa lama keessa darbu barbaaduuf itti fayyadamaa.

Of eeggadhu

Tarree duraa keessatti x fi y akka jijjiiramaatti tursiisaa. Gatii lakkoofsaa bakka isaanii hin galchinaa.

Grade 11 Mathematics, fuula 207-250 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Garagaltoo Maatrikisii Adjoint FayyadamuunFoormulaa

A^{-1} = \frac{1}{\det(A)} \text{Adj}(A)

A^{-1} = \frac{1}{\det(A)} \text{Adj}(A)

$A^{-1}$: Garagaltoo maatrikisii iskweerii A
$\det(A)$: Determinantii maatrikisii A, zeeroo kan hin taane
$\text{Adj}(A)$: Maatrikisii adjoint, tiraansipoozii maatrikisii kofaaktarii

Yoom akka itti fayyadamtu

Garagaltoo maatrikisii kofaaktarootaa fi determinantii fayyadamuun shallaguuf itti fayyadamaa.

Of eeggadhu

Maatrikisiin adjoint tiraansipoozii maatrikisii kofaaktariiti, kanaaf tarree fi toora waljijjiiruu hin dagatinaa.

Grade 11 Mathematics, fuula 207-250 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Gaaffii shaakalaa boqonnaa 4

Gaaffii filannoo 10 boqonnaa kana irratti barreeffaman, tokkoon tokkoon isaanii deebii fi sababa qabu. Deebii banuu dura yaali. Gaaffiin kun Ingiliffaa tiin.

Gaaffii 1

What is another name for an invertible matrix?

A.A singular matrix
B.A nonsingular matrix
C.A zero matrix
D.A row matrix

Deebii fi sababa agarsiisi

Deebii: B A nonsingular matrix

Sababa

By definition, an invertible matrix is also called a nonsingular matrix, while a matrix that is not invertible is called a singular matrix.

Gaaffii 2

If A and B are invertible matrices of the same size, which formula correctly gives the inverse of their product, (AB)^-1?

A.A^-1 B^-1
B.B^-1 A^-1
C.(BA)^-1
D.A^t B^t

Deebii fi sababa agarsiisi

Deebii: B B^-1 A^-1

Sababa

According to the properties of invertible matrices, the inverse of the product is reversed: (AB)^-1 = B^-1A^-1.

Gaaffii 3

Which term describes a diagonal matrix in which all main diagonal elements are equal?

A.Scalar matrix
B.Column matrix
C.Upper triangular matrix
D.Zero matrix

Deebii fi sababa agarsiisi

Deebii: A Scalar matrix

Sababa

A diagonal matrix where all elements along the main diagonal are equal is called a scalar matrix.

Gaaffii 4

Which of the following is NOT one of the three standard elementary row operations?

A.Swapping two rows
B.Multiplying a row by a non-zero constant
C.Adding a constant multiple of one row to another row
D.Squaring all entries in a single row

Deebii fi sababa agarsiisi

Deebii: D Squaring all entries in a single row

Sababa

The three elementary row operations are swapping rows, scaling a row by a non-zero constant, and pivoting (adding a multiple of one row to another). Squaring entries is not an elementary row operation.

Gaaffii 5

Given the matrix A = [[1, 1], [1, 2]], what is its inverse A^-1?

A.[[2, -1], [-1, 1]]
B.[[-2, 1], [1, -1]]
C.[[1, -1], [-1, 2]]
D.[[2, 1], [1, 1]]

Deebii fi sababa agarsiisi

Deebii: A [[2, -1], [-1, 1]]

Sababa

Using elementary row operations to reduce (A|I₂), we find that A^-1 = [[2, -1], [-1, 1]].

Gaaffii 6

If the inverse of matrix A is A^-1 = [[1, 1], [1, 2]], what is the inverse of its transpose, (A^t)^-1?

A.[[2, -1], [-1, 1]]
B.[[1, 1], [1, 2]]
C.[[1, 2], [1, 1]]
D.[[2, 1], [1, 1]]

Deebii fi sababa agarsiisi

Deebii: B [[1, 1], [1, 2]]

Sababa

By the properties of invertible matrices, (A^t)^-1 = (A^-1)^t. Transposing A^-1 = [[1, 1], [1, 2]] gives [[1, 1], [1, 2]].

Gaaffii 7

Why is the matrix B = [[1, 2], [2, 4]] not invertible?

A.It is not a square matrix.
B.Its entries are all positive numbers.
C.Row reduction of (B|I_2) produces a row of zeros in the left matrix, so it cannot reduce to I_2.
D.Its transpose is equal to itself.

Deebii fi sababa agarsiisi

Deebii: C Row reduction of (B|I_2) produces a row of zeros in the left matrix, so it cannot reduce to I_2.

Sababa

Applying the row operation R₂ → R₂ - 2R₁ transforms the left matrix to have a bottom row of all zeros. Since it cannot be reduced to the identity matrix I₂, matrix B is singular (not invertible).

Gaaffii 8

A laboratory technician wants to mix a 30% salt solution with a 20% salt solution to obtain 500 liters of a 25% salt solution. How many liters of each solution are needed?

A.200 liters of 30% and 300 liters of 20%
B.250 liters of 30% and 250 liters of 20%
C.150 liters of 30% and 350 liters of 20%
D.100 liters of 30% and 400 liters of 20%

Deebii fi sababa agarsiisi

Deebii: B 250 liters of 30% and 250 liters of 20%

Sababa

Setting up the system: x + y = 500 and 0.3x + 0.2y = 125. Solving this system gives y = 250 liters and x = 250 liters.

Gaaffii 9

Given the 3x3 matrix C = [[1, 1, 1], [0, 1, 2], [0, 0, 1]], what is its inverse C^-1?

A.[[1, -1, 1], [0, 1, -2], [0, 0, 1]]
B.[[1, 1, -1], [0, 1, 2], [0, 0, 1]]
C.[[1, 0, 0], [-1, 1, 0], [1, -2, 1]]
D.[[-1, 1, -1], [0, -1, 2], [0, 0, -1]]

Deebii fi sababa agarsiisi

Deebii: A [[1, -1, 1], [0, 1, -2], [0, 0, 1]]

Sababa

By augmenting with I₃ and applying row operations (R₁ → R₁ - R₂ and R₂ → R₂ - 2R₃), we obtain C^-1 = [[1, -1, 1], [0, 1, -2], [0, 0, 1]].

Gaaffii 10

In a triangle, the smallest angle measures 10 degrees more than half of the largest angle. The middle angle measures 12 degrees more than the smallest angle. What are the measures of the three angles?

A.40 degrees, 52 degrees, and 88 degrees
B.47 degrees, 59 degrees, and 74 degrees
C.45 degrees, 57 degrees, and 78 degrees
D.50 degrees, 62 degrees, and 68 degrees

Deebii fi sababa agarsiisi

Deebii: B 47 degrees, 59 degrees, and 74 degrees

Sababa

Let x, y, z be the angles. The equations are x + y + z = 180, x - 0.5z = 10, and -x + y = 12. Solving using row operations yields z = 74 degrees, y = 59 degrees, and x = 47 degrees.